A New Class of Doubly Nonlinear Evolution Equations M. ASO, T. FUKAO and N. KENMOCHI Technical Reports of Mathematical Sciences Chiba University, Vol. 18 (2002), No. 13

A new class of doubly nonlinear evolution equations

M. Aso, T. Fukao and N. Kenmochi

Abstract. In this paper we consider the solvability for a new class of doubly nonlinear evolution equations. The motivation of this work comes from a transmission problem of two degenerate parabolic equations with convection term, in which the transmission boundary is time-dependent. We give an abstract existence result, and show that the weak variational formulation for the transmission problem can be solved by applying this abstract result. In our existence proof, the abstract theory of pseudo-monotone operators is useful.

1 Introduction

Let 0 < T < ∞. We consider an evolution equation of the form:

u′(t) + K(t, θ(t)) + G(t, u(t)) = f(t) in V^{∗} for a.e. t ∈ [0, T],  (1.1)
u(0) = u_{0},  (1.2)

where u′ := du/dt. Here, for each t ∈ [0, T], K(t, ·) is a weakly continuous operator in a reflexive Banach space V, G(t, ·) is a weakly continuous operator in a Hilbert space H, where V is imbedded densely and compactly in H, f is a given source function and u_{0} is an initial datum. Equation (1.1) is considered with the following relation between θ and u:

θ(t) = ∂ψ^{t}(u(t)) in V for a.e. t ∈ [0, T],  (1.3)

where {ψ^{t}} is a family of proper, lower semicontinuous and convex functions on the dual space V^{∗} of V and ∂ψ^{t} is the subdifferential of ψ^{t} from V^{∗} into V; especially in our setting the subdifferential ∂ψ^{t} is assumed to be singlevalued.

Some types of doubly nonlinear evolution equations have been considered so far, for example,

A(u′(t)) + B(u(t)) ∋ f(t),  (1.4)

which was considered by Arai [1], Senba [20], Colli and Visintin [8] and Colli [7], where A and B are maximal monotone, possibly nonlinear and multivalued operators from V into V^{∗}; in Colli [7], A is bounded and B is unbounded so that the domain D(B) is contained in a Banach space W imbedded compactly in V. Another type of doubly nonlinear evolution equations is of the form:

\frac{d}{dt}A(u(t)) + B(u(t)) ∋ f(t),  (1.5)

which was treated, for instance, by Kenmochi [15], Kenmochi and Pawłow [16], [17], and recently Maitre and Witomski [18]. In our problem, the unknown u can be eliminated by (1.3). In fact, using the conjugate convex function ψ^{t∗} of ψ^{t}, we obtain from (1.3) that

u(t) ∈ ∂ψ^{t∗}(θ(t)) in V^{∗},  (1.6)

where ∂ψ^{t∗} is the subdifferential of ψ^{t∗} from V into V^{∗}. Hence (1.1) is formally written in the form

\frac{d}{dt}∂ψ^{t∗}(θ(t)) + K(t, θ(t)) + G(t, ∂ψ^{t∗}(θ(t))) ∋ f(t) in V^{∗}.  (1.7)

This is a new type of doubly nonlinear evolution equations in respect that the time derivative of ∂ψ^{t∗}(θ(t)) is included in the equation as well as a highly nonlinear perturbation G(t, ∂ψ^{t∗}(θ(t))). We have not noticed any results on this class of evolution equations. In this paper we give an existence result for problem {(1.1)-(1.3)}. In our construction of a solution, we approximate (1.1) by a time discretization scheme. After getting some uniform estimates we discuss its convergences to obtain a solution of our problem.

The above type of evolution equations arises from transmission problems of two degenerate parabolic equations. This problem has been studied by Fukao, Kenmochi and Pawłow [11]. In their paper it was treated as a system of transmission-Stefan type. Two degenerate parabolic equations are combined by the transmission condition. In this case the problem is formulated as the initial value problem for (1.1) in an abstract Banach space, which will be discussed in detail in the last section.

2 Main Result

Throughout this paper, we denote by V a real reflexive Banach space with norm | · |_{V}, by V^{∗} the dual space of V and by ⟨·, ·⟩_{V^{∗},V} the duality pairing between V^{∗} and V. Moreover, let H be a real Hilbert space with inner product (·, ·)_{H} and norm | · |_{H} such that the following dense and compact imbeddings are satisfied:

V ↪^{d} H ↪^{d} V^{∗}.

Without loss of generality we may assume that V and V^{∗} are strictly convex spaces. Now consider the initial value problem (DN):= {(2.1)-(2.3)}:

u′(t) + K(t, θ(t)) + G(t, u(t)) = f(t) in V^{∗} for a.e. t ∈ [0, T],  (2.1)
θ(t) = ∂ψ^{t}(u(t)) in V for a.e. t ∈ [0, T],  (2.2)
u(0) = u_{0},  (2.3)

under the following assumptions (A1)-(A3):

(A1) For each t ∈ [0, T], ψ^{t} is a proper, lower semicontinuous and convex function on V^{∗} with D(ψ^{t}) ⊂ H, and ψ^{t} is coercive, that is, there exists a positive constant C_{0}, independent of t, such that

ψ^{t}(u) ≥ C_{0}|u|_{H}^{2} for all u ∈ D(ψ^{t}).  (2.4)

Moreover ψ^{t} satisfies the following time-dependent condition: For each t, s ∈ [0, T] and u ∈ D(ψ^{s}), there exists ũ ∈ D(ψ^{t}) such that

|ũ − u|_{V^{∗}} ≤ C_{1}|t − s|(ψ^{s}(u)^{\frac{1}{2}} + 1),  (2.5)

ψ^{t}(ũ) − ψ^{s}(u) ≤ C_{1}|t − s|(ψ^{s}(u) + 1),  (2.6)

where C_{1} is a positive constant independent of t, s and u. In Addition, ψ^{t} is strongly monotone in the following sense:

⟨u_{1} − u_{2}, θ_{1} − θ_{2}⟩_{V^{∗},V} ≥ C_{2}|θ_{1} − θ_{2}|_{H}^{2} for all θ_{i} = ∂ψ^{t}(u_{i}) and i = 1, 2. (2.7)

(A2) The operator K(t, ·) : D(K(t, ·)) = V → V^{∗} satisfies that

(coerciveness)

⟨K(t, θ), θ⟩_{V^{∗},V} ≥ C_{3}|θ|_{V}^{2} − C_{4} for all θ ∈ V and t ∈ [0, T],  (2.8)

(boundedness)

|K(t, θ)|_{V^{∗}} ≤ C_{5}|θ|_{V} + C_{6} for all θ ∈ V and t ∈ [0, T],  (2.9)

where C_{3}, C_{4}, C_{5} and C_{6} are positive constants. Moreover K(·, ·) satisfies the following property: If θ_{n} → θ in L^{2}(0, T; H) and weakly in L^{2}(0, T; V) as n → +∞, then

K(·, θ_{n}(·, ·)) → K(·, θ(·, ·)) weakly in L^{2}(0, T; V^{∗}) as n → +∞.  (2.10)

In Addition, for each θ ∈ V, K(·, θ) is continuous with respect to t in the following sense: There exists a continuous function ω : [0, +∞) → [0, +∞) with ω(0) = 0 such that

|K(t, θ) − K(s, θ)|_{V^{∗}} ≤ ω(|t − s|)(|θ|_{V} + 1) for all t, s ∈ [0, T].  (2.11)

(A3) The operator G(t, ·) : D(G(t, ·)) = H → V^{∗} satisfies that

(boundedness)

|G(t, u)|_{V^{∗}} ≤ C_{7}|u|_{H} + C_{8} for all u ∈ H and t ∈ [0, T],  (2.12)

where C_{7} and C_{8} are positive constants. Moreover G(·, ·) satisfies the following property: If t_{n} → t and u_{n} → u weakly in H as n → +∞, then

G(t_{n}, u_{n}) → G(t, u) weakly in V^{∗} as n → +∞.  (2.13)

Here we give the definition of a solution of (DN).

Definition 2.1 A pair {u, θ} of functions u, θ : [0, T] → H is called a solution of (DN), if u ∈ W^{1,2}(0, T; V^{∗}) ∩ L^{∞}(0, T; H), θ ∈ L^{2}(0, T; V) and {u, θ} satisfies (2.1), (2.2) and (2.3).

Our main theorem is formulated now:

Theorem 2.1 Assume that (A1)-(A3) hold. Given f ∈ L^{2}(0, T; V^{∗}) and u_{0} ∈ D(ψ^{0}), (DN) has at least one solution {u, θ}.

We shall prove our existence theorem in sections 3 and 4. Our main idea for the construction of a solution is to employ the time discretization method for (DN).

3 Approximation of (DN)

In order to construct a solution of (DN), we use the time discretization method to approximate problem (DN). For an arbitrary N ∈ N, we put h_{N} := T/N, t_{k}^{N} := kh_{N} for k = 0, 1, …, N and u_{0}^{N} := u_{0}. Then, we consider the following time discretization scheme for (DN): Find a pair {u_{k}^{N}, θ_{k}^{N}} of functions satisfying that

\frac{u_{k}^{N} − u_{k−1}^{N}}{h_{N}} + K(t_{k}^{N}, θ_{k}^{N}) = −G(t_{k−1}^{N}, u_{k−1}^{N}) + f_{k−1}^{N} in V^{∗},  (3.1)

with

θ_{k}^{N} = ∂ψ^{t_{k}^{N}} (u_{k}^{N}) in V for k = 1, 2, …, N,  (3.2)

where f_{k}^{N} is the discrete approximation of f given by

f_{k}^{N} := \frac{1}{h_{N}} ∫_{t_{k}^{N}}^{t_{k+1}^{N}} f(s)ds for k = 0, 1, …, N − 1.  (3.3)

Now we discuss the solvability of (DN)_{N} := {(3.1), (3.2)}. For simplicity, put A(t, ·) := (∂ψ^{t})^{−1}. Note that (DN)_{N} can be written in the form:

A(t_{k}^{N}, θ_{k}^{N}) + h_{N}K(t_{k}^{N}, θ_{k}^{N}) ∋ u_{k−1}^{N} − h_{N}G(t_{k−1}^{N}, u_{k−1}^{N}) + h_{N}f_{k−1}^{N} in V^{∗}.  (3.4)

In order to construct the time discretization scheme, for each fixed t_{k}^{N}, it suffices to check the existence of a solution θ ∈ V of

A(t_{k}^{N}, θ) + h_{N}K(t_{k}^{N}, θ) ∋ g^{∗} in V^{∗},  (3.5)

where g^{∗} is given in V^{∗}. Now we recall the following general theory of nonlinear operators.

Proposition 3.1 Let Z be a real reflexive Banach space and Z^{∗} be the dual space of Z. Assume that a multivalued operator N : Z → Z^{∗} satisfies the following conditions (N1) and (N2):

(N1) For each z ∈ Z, N z is a non-empty, bounded, convex and closed set in V^{∗}.

(N2) N is weakly sequentially upper semicontinuous and it is coercive, that is,

lim_{|z|_{Z}→+∞} inf_{z^{∗}∈N z} \frac{⟨z^{∗}, z⟩_{Z^{∗},Z}}{|z|_{Z}} = +∞.

Then N is surjective, that is, R(N) = Z^{∗}.

This proposition is well-known; for instance, it is an immediate consequence of the abstract results on the surjectiveness of nonlinear multivalued operators of Type M (see Brézis [3], Browder and Hess [6], Kenmochi [14] and so on).

Lemma 3.1 (i) Let t ∈ [0, T] and N ∈ N. Then D(A(t, ·)) = V and A(t, θ)+h_{N}K(t, θ) is a non-empty, bounded, convex and closed set in V^{∗} for each θ ∈ V.

(ii) A(t, ·) + h_{N}K(t, ·) is weakly sequentially upper semicontinuous and coercive as a mapping from V into V^{∗}. Proof. (i) Since A(t, ·)^{−1} = ∂ψ^{t}, by (2.4) we have R(A^{−1}(t, ·)) = D(A(t, ·)) = V and the boundedness of A(t, ·). From the maximal monotonicity of A(t, ·) it follows immediately that A(t, θ) + h_{N}K(t, θ) is non-empty, bounded, convex and closed for all θ ∈ V. (ii) The coerciveness of A(t, ·)+h_{N}K(t, ·) is seen from (2.8) and the monotonicity of A(t, ·). Lastly we check that A(t, ·) + h_{N}K(t, ·) is weakly sequentially upper semicontinuous in V^{∗}. Now note that K(t, ·) is a singlevalued and weakly continuous operator from V into V^{∗}, and A(t, ·) is a subdifferential operator with R(A(t, ·)) = D(A^{−1}(t, ·)) ⊂ D(ψ^{t}) ⊂ H. Therefore, by the compact imbedding V ↪ H, we see that the graph G(A(t, ·)+h_{N}K(t, ·)) is weakly closed in V × V^{∗}, which implies the conclusion. □

This lemma and Proposition 3.1 imply that for each fixed N ∈ N, k = 0, 1, …, N, and any given g^{∗} in V^{∗}, there exists θ ∈ V which satisfies (3.5). In our setting, the subdifferential operator ∂ψ^{t} is singlevalued. Thus, Lemma 3.1 and Proposition 3.1 show that our approximation scheme (DN)_{N} has solutions u_{k}^{N} ∈ H and θ_{k}^{N} ∈ V for each N ∈ N and k = 0, 1, …, N.

Moreover we obtain the following estimates for {u_{k}^{N}, θ_{k}^{N}}.

Lemma 3.2 There exists a positive constant M_{1} independent of N such that

ψ^{t_{k}^{N}} (u_{k}^{N}) ≤ M_{1} for all k = 0, 1, …, N,  (3.6)
h_{N} ∑_{k=1}^{N} |θ_{k}^{N}|_{V}^{2} ≤ M_{1}.  (3.7)

Proof. Multiplying (DN)_{N} by θ_{k}^{N} we obtain

\frac{1}{h_{N}} (u_{k}^{N}, θ_{k}^{N})_{H} − \frac{1}{h_{N}} (u_{k−1}^{N}, θ_{k}^{N})_{H} + ⟨K(t_{k}^{N}, θ_{k}^{N}), θ_{k}^{N}⟩_{V^{∗},V}

= −⟨G(t_{k−1}^{N}, u_{k−1}^{N}), θ_{k}^{N}⟩_{V^{∗},V} + ⟨f_{k−1}^{N}, θ_{k}^{N}⟩_{V^{∗},V} .  (3.8)

By conditions (2.5) and (2.6) in (A1) for t := t_{k}^{N}, s := t_{k−1}^{N} and u := u_{k−1}^{N} ∈ D(ψ^{t_{k−1}^{N}}) we can find an element ũ_{k}^{N} ∈ D(ψ^{t_{k}^{N}}) such that

|ũ_{k}^{N} − u_{k−1}^{N}|_{V^{∗}} ≤ C_{1}h_{N} (ψ^{t_{k−1}^{N}}(u_{k−1}^{N})^{\frac{1}{2}} + 1),  (3.9)
ψ^{t_{k}^{N}} (ũ_{k}^{N}) − ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) ≤ C_{1}h_{N}(ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) + 1).  (3.10)

By the definition of subdifferential ∂ψ^{t_{k}^{N}} and (3.9) we get

\frac{1}{h_{N}} (u_{k}^{N}, θ_{k}^{N})_{H} − \frac{1}{h_{N}} (u_{k−1}^{N}, θ_{k}^{N})_{H}

= \frac{1}{h_{N}} (u_{k}^{N} − ũ_{k}^{N}, θ_{k}^{N})_{H} + \frac{1}{h_{N}} (ũ_{k}^{N} − u_{k−1}^{N}, θ_{k}^{N})_{H}
≥ \frac{1}{h_{N}} (ψ^{t_{k}^{N}} (u_{k}^{N}) − ψ^{t_{k}^{N}} (ũ_{k}^{N})) − C_{1} (ψ^{t_{k−1}^{N}}(u_{k−1}^{N})^{\frac{1}{2}} + 1) |θ_{k}^{N}|_{V} .

Moreover, by Schwarz’s inequality and (3.10),

\frac{1}{h_{N}} (ψ^{t_{k}^{N}} (u_{k}^{N}) − ψ^{t_{k}^{N}} (ũ_{k}^{N})) − C_{1} (ψ^{t_{k−1}^{N}}(u_{k−1}^{N})^{\frac{1}{2}} + 1) |θ_{k}^{N}|_{V}

≥ \frac{1}{h_{N}} (ψ^{t_{k}^{N}} (u_{k}^{N}) − ψ^{t_{k−1}^{N}}(u_{k−1}^{N})) + \frac{1}{h_{N}} (ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) − ψ^{t_{k}^{N}} (ũ_{k}^{N}))
− C_{1}^{2}C_{ε_{1}}ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) − ε_{1}|θ_{k}^{N}|_{V}^{2} − C_{1}^{2}C_{ε_{1}} − ε_{1}|θ_{k}^{N}|_{V}^{2}
≥ \frac{1}{h_{N}} (ψ^{t_{k}^{N}} (u_{k}^{N}) − ψ^{t_{k−1}^{N}}(u_{k−1}^{N})) − C_{1}(ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) + 1)
− 2ε_{1}|θ_{k}^{N}|_{V}^{2} − C_{1}^{2}C_{ε_{1}}ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) − C_{1}^{2}C_{ε_{1}}
= \frac{1}{h_{N}} (ψ^{t_{k}^{N}} (u_{k}^{N}) − ψ^{t_{k−1}^{N}}(u_{k−1}^{N})) − 2ε_{1}|θ_{k}^{N}|_{V}^{2} − (C_{1} + C_{1}^{2}C_{ε_{1}})ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) − (C_{1} + C_{1}^{2}C_{ε_{1}});

hereafter ε_{i} is an arbitrary positive constant and C_{ε_{i}} := 1/(4ε_{i}) for each i ∈ N. Next, by the coerciveness (2.8) of K we have

⟨K(t_{k}^{N}, θ_{k}^{N}), θ_{k}^{N}⟩_{V^{∗},V} ≥ C_{3}|θ_{k}^{N}|_{V}^{2} − C_{4}.

By the boundedness (2.12) of G we have

⟨−G(t_{k−1}^{N}, u_{k−1}^{N}), θ_{k}^{N}⟩_{V^{∗},V} +⟨f_{k−1}^{N}, θ_{k}^{N}⟩_{V^{∗},V} ≤ C_{7}^{2}C_{ε_{2}}|u_{k−1}^{N}|_{H}^{2}+C_{8}^{2}C_{ε_{2}}+3ε_{2}|θ_{k}^{N}|_{V}^{2} +C_{ε_{2}}|f_{k−1}^{N}|_{V^{∗}}^{2}.

We obtain from (3.8) with the help of (2.4) that

\frac{1}{h_{N}} (ψ^{t_{k}^{N}} (u_{k}^{N}) − ψ^{t_{k−1}^{N}}(u_{k−1}^{N})) + (C_{3} − 2ε_{1} − 3ε_{2})|θ_{k}^{N}|_{V}^{2}

≤ (C_{1} + C_{1}^{2}C_{ε_{1}} + \frac{C_{7}^{2}C_{ε_{2}}}{C_{0}} ) ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) + C_{ε_{2}}|f_{k−1}^{N}|_{V^{∗}}^{2} + (C_{1} + C_{1}^{2}C_{ε_{1}} + C_{4} + C_{8}^{2}C_{ε_{2}}),

(3.11)

where ε_{1} and ε_{2} are chosen so as to satisfy C_{3} − 2ε_{1} − 3ε_{2} > 0. This inequality can be written in the form

ψ^{t_{k}^{N}} (u_{k}^{N}) + M_{2} ≤ (1 + \frac{T M_{3}}{N} ) (ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) + M_{2}) for all k = 1, …, N,  (3.12)

where M_{2} and M_{3} are positive constants independent of N; M_{2} depends on |f|_{L^{2}(0,T;V^{∗})} and M_{3} := C_{1} + C_{1}^{2}C_{ε_{1}} + (C_{7}^{2}C_{ε_{2}}/C_{0}). Here, applying the discrete Gronwall’s inequality to (3.12), we obtain

ψ^{t_{k}^{N}} (u_{k}^{N}) ≤ exp(1 + T M_{3})(ψ^{0}(u_{0}) + M_{2}) for all k = 1, 2, …, N.  (3.13)

Next, we multiply (3.11) and (3.13) by h_{N} and sum them up for k = 1, 2, …, N to obtain

ψ^{T} (u_{N}^{N}) + (C_{3} − 2ε_{1} − 3ε_{2})h_{N} ∑_{k=1}^{N} |θ_{k}^{N}|_{V}^{2} ≤ M_{3}h_{N} ∑_{k=1}^{N} ψ^{t_{k−1}^{N}}(u_{k−1}^{N}) + M_{2}T + ψ^{0}(u_{0}).

Hence (3.7) is obtained from this inequality together with (3.13). □

Now, let {u_{i}^{N}, θ_{i}^{N}}_{i=0}^{N} be the discretization scheme of (3.1), (3.2) and (3.3) constructed above and define the piecewise linear H-valued function u^{N} generated by {u_{i}^{N}} on [0, T], that is, u^{N}(0) := u_{0} and

u^{N}(t) := \frac{(t − t_{k−1}^{N})u_{k}^{N} + (t_{k}^{N} − t)u_{k−1}^{N}}{h_{N}} if t ∈ (t_{k−1}^{N}, t_{k}^{N}] for k = 1, 2, …, N.

It is clear that u^{N} ∈ C([0, T]; H) for each N ∈ N. Moreover, we have the following lemma:

Lemma 3.3 {u^{N}}_{N=1}^{∞} is equicontinuous as a family of functions from [0, T] into V^{∗} and uniformly bounded on [0, T] as H-valued functions. Proof. Using Lemma 3.2, we have |u_{k}^{N}|_{H} ≤ (M_{1}/C_{0})^{1/2} for k = 1, 2, …, N, so that {u^{N}} is uniformly bounded on [0, T] as H-valued functions. Next we take s, t ∈ [0, T] with s ≤ t such that s ∈ [t_{k−1}^{N}, t_{k}^{N}] and t ∈ [t_{j−1}^{N}, t_{j}^{N}], where 1 ≤ k ≤ j ≤ N. We see from the boundedness (2.9) of K and (2.12) of G together with (2.4), (3.5) and (3.6) that there exists a positive constant M_{4}, which depends on C_{6} and C_{8}, such that

|u^{N}(t) − u^{N}(s)|_{V^{∗}}

≤ 2 ∑_{ℓ=k−1}^{j} |u_{ℓ}^{N} − u_{ℓ−1}^{N}|_{V^{∗}}

≤ 2 (h_{N} ∑_{ℓ=k−1}^{j} |K(t_{ℓ}^{N}, θ_{ℓ}^{N})|_{V^{∗}} + h_{N} ∑_{ℓ=k−1}^{j} |G(t_{ℓ}^{N}, u_{ℓ}^{N})|_{V^{∗}} + h_{N} ∑_{ℓ=k−1}^{j} |f_{ℓ}^{N}|_{V^{∗}} )

≤ 2(h_{N}(j − k + 1))^{\frac{1}{2}} \{C_{5}(h_{N}∑_{k=1}^{N} |θ_{k}^{N}|_{V}^{2} )^{\frac{1}{2}}+ C_{7}(h_{N}∑_{k=1}^{N} \frac{ψ^{t_{k}^{N}} (u_{k}^{N})}{C_{0}})^{\frac{1}{2}}+ M_{4} + |f|_{L^{2}(0,T;V^{∗})}\}

≤ 2(|t − s| + 2h_{N})^{\frac{1}{2}} \{C_{5}M1^{\frac{1}{2}} + C_{7} (\frac{T M_{1}}{C_{0}} )^{\frac{1}{2}} + M_{4} + |f|_{L^{2}(0,T;V^{∗})}\} ,

which shows the equicontinuity of {u^{N}} on [0, T]. □

By virtue of Lemmas 3.2, 3.3 and the Ascoli-Arzela’s theorem it is easily seen that there exists a subsequence {N_{n}} ⊂ {N} and there exists a function u ∈ C([0, T]; V^{∗}) such that

u^{N_{n}} → u in C([0, T]; V^{∗}) as n → +∞.  (3.14)

Moreover, define the V^{∗}-valued step function ū^{N} and the V -valued step function θ̄^{N} on [0, T] by putting ū^{N}(0) := u_{0}, θ̄^{N}(0) := ∂ψ^{0}(u_{0}) and

ū^{N}(t) := u_{k}^{N}, θ̄^{N}(t) := θ_{k}^{N} if t ∈ (t_{k−1}^{N}, t_{k}^{N}] for k = 1, 2, …, N.  (3.15)

Then (3.14) implies

sup_{t∈[0,T]} |ū^{N_{n}}(t) − u(t)|_{V^{∗}} → 0 as n → +∞.  (3.16)

Additionally, we obtain the following convergence and relation between u and θ:

Lemma 3.4 There exists a subsequence {N_{m}} ⊂ {N_{n}} with N_{m} → ∞ as m → +∞ and there exists a function θ ∈ L^{2}(0, T; V) such that

θ̄^{N_{m}} → θ weakly in L^{2}(0, T; V) as m → +∞.  (3.17)

Moreover the pair {u, θ} satisfies that θ(t) = ∂ψ^{t}(u(t)) in V for a.e. t ∈ [0, T]. Proof. By (3.7) in Lemma 3.2, we see that {θ̄^{N_{n}}}_{n=1}^{∞} = {∂ψ^{t}(ū^{N_{n}})}_{n=1}^{∞} is bounded in L^{2}(0, T; V), which implies (3.17) for some subsequence {N_{m}}. Next, we show that θ(t) = ψ^{t}(u(t)) for a.e. t ∈ [0, T] which is equivalent to

∫_{0}^{T} ⟨η(t) − u(t), θ(t)⟩_{V^{∗},V} dt ≤ ∫_{0}^{T} ψ^{t}(η(t))dt − ∫_{0}^{T} ψ^{t}(u(t))dt for all η ∈ L^{2}(0, T; V^{∗}).

(3.18)

Let η be any function in L^{2}(0, T; V^{∗}) such that ∫_{0}^{T} ψ^{t}(η(t))dt < +∞. Then there are step functions η̄^{N_{m}} ∈ L^{2}(0, T; V^{∗}) for all m ∈ N such that

η̄^{N_{m}} → η in L^{2}(0, T; V^{∗}) as m → +∞,  (3.19)
lim sup_{m→+∞} ∑_{k=1}^{N_{m}} ∫_{t_{k−1}^{Nm}}^{t_{k}^{Nm}} ψ^{t_{k}^{Nm}}(η̄^{N_{m}}(t))dt ≤ ∫_{0}^{T} ψ^{t}(η(t))dt;  (3.20)

see Kenmochi [15]. Now, from the definition of subdifferential it follows that

∑_{k=1}^{N_{m}} ∫_{t_{k−1}^{Nm}}^{t_{k}^{Nm}}⟨η̄^{N_{m}}(t) − ū^{N_{m}}(t), θ̄^{N_{m}}(t)⟩_{V^{∗},V} dt

≤ ∑_{k=1}^{N_{m}} ∫_{t_{k−1}^{Nm}}^{t_{k}^{Nm}} ψ^{t_{k}^{Nm}}(η̄^{N_{m}}(t))dt − ∑_{k=1}^{N_{m}} ∫_{t_{k−1}^{Nm}}^{t_{k}^{Nm}} ψ^{t_{k}^{Nm}}(ū^{N_{m}}(t))dt.  (3.21)

Let t be any time in (0, T]. By virtue of (2.5) and (2.6) in (A1) for t ∈ (t_{k−1}^{N_{m}}, t_{k}^{N_{m}}], s := t_{k}^{N_{m}} and ū^{N_{m}}(t) ∈ D(ψ^{t_{k}^{Nm}}) there exists a function z̃_{m} ∈ D(ψ^{t}) such that

|z̃_{m} − ū^{N_{m}}(t_{k}^{N_{m}})|_{V^{∗}} ≤ C_{1}|t − t_{k}^{N_{m}}|(ψ^{t_{k}^{Nm}}(ū^{N_{m}}(t_{k}^{N_{m}}))^{\frac{1}{2}} + 1),  (3.22)
ψ^{t}(z̃_{m}) − ψ^{t_{k}^{Nm}}(ū^{N_{m}}(t_{k}^{N_{m}})) ≤ C_{1}|t − t_{k}^{N_{m}}|(ψ^{t_{k}^{Nm}}(ū^{N_{m}}(t_{k}^{N_{m}})) + 1).  (3.23)

We now observe from (3.16) and (3.22) that

z̃_{m} → u(t) in V^{∗} as m → +∞.  (3.24)

On the other hand, using (3.23), (3.24) and the fact that ψ^{t} is lower semicontinuous on V^{∗}, we see

lim inf_{m→+∞} ψ^{t_{k}^{Nm}}(ū^{N_{m}}(t_{k}^{N_{m}})) ≥ lim inf_{m→+∞} (\frac{1}{1 + C_{1}|t − t_{k}^{N_{m}}|}ψ^{t}(z̃_{m}) − \frac{C_{1}|t − t_{k}^{N_{m}}|}{1 + C_{1}|t − t_{k}^{N_{m}}|})
≥ ψ^{t}(u(t)).

Thus, taking the limit as m → +∞ in (3.21) and using (3.16), (3.17), (3.19) and (3.20) with the help of Fatou’s lemma, we get (3.18). □

4 Proof of theorem

In this section we prove the main theorem after preparing two lemmas with some lemmas. Let {ū^{N_{m}}, θ̄^{N_{m}}}_{m=1}^{∞} and {u, θ} be such as in the last section.

Lemma 4.1 Let {θ_{j}} be any sequence in C([0, T]; V) such that θ_{j} → θ in L^{2}(0, T; V) as j → +∞. Put

u_{j}(t) := (∂ψ^{t} + F^{−1})^{−1}(θ_{j}(t) + F^{−1}u(t)) in V^{∗} for all t ∈ [0, T].  (4.1)

where F is the duality mapping from V to V^{∗}. Then u_{j} ∈ C([0, T]; V^{∗}) and

u_{j} → u weakly in L^{2}(0, T; V^{∗}) as j → +∞.  (4.2)

Proof. First note that ∂ψ^{t} + F^{−1} is a singlevalued and surjective operator from V^{∗} into V as well as from L^{2}(0, T; V^{∗}) into L^{2}(0, T; V). Clearly u_{j} ∈ L^{2}(0, T; V^{∗}). We show u_{j} ∈ C([0, T]; V^{∗}) as follows: For each fixed j ∈ N and t ∈ [0, T], let {t_{n}} ⊂ [0, T] be any sequence such that t_{n} → t as n → +∞. Our claim is to show that

u_{j}(t_{n}) → u_{j}(t) in V^{∗} as n → +∞.  (4.3)

Take a function w ∈ C([0, T]; V^{∗}) with w(t) ∈ D(ψ^{t}) such that

t → ψ^{t}(w(t)) is bounded in [0, T],

for example, by the existence result in Kenmochi [15] we can find w as a solution of the Cauchy problem

\frac{∂w}{∂t}(t) + ∂_{H}ψ^{t}(w(t)) = 0 in H,

w(0) = 0 in H,

where ∂_{H}ψ^{t} is the subdifferential of ψ^{t} in H. Put g_{j}(t) := θ_{j}(t) + F^{−1}u(t) in V for all t ∈ [0, T]. Then, from the definition of subdifferential and assumption (2.4) in (A1) it follows that

⟨u_{j}(t_{n}) − w(t_{n}), g_{j}(t_{n})⟩_{V^{∗},V}
= ⟨u_{j}(t_{n}) − w(t_{n}), ∂ψ^{t_{n}}(u_{j}(t_{n})) + F^{−1}u_{j}(t_{n})⟩_{V^{∗},V}
≥ ψ^{t_{n}}(u_{j}(t_{n})) − ψ^{t_{n}}(w(t_{n})) + |u_{j}(t_{n})|_{V^{∗}}^{2} − |w(t_{n})|_{V^{∗}}|u_{j}(t_{n})|_{V^{∗}}
≥ C_{0}|u_{j}(t_{n})|_{H}^{2} − ψ^{t_{n}}(w(t_{n})) + |u_{j}(t_{n})|_{V^{∗}}^{2} − \frac{1}{2}|w(t_{n})|_{V^{∗}}^{2} − \frac{1}{2}|u_{j}(t_{n})|_{V^{∗}}^{2}
≥ C_{0}|u_{j}(t_{n})|_{H}^{2} − ψ^{t_{n}}(w(t_{n})) − \frac{1}{2}|w(t_{n})|_{V^{∗}}^{2}.

On the other hand, there exists a positive constant M_{5} such that

⟨u_{j}(t_{n}) − w(t_{n}), g_{j}(t_{n})⟩_{V^{∗},V} ≤ M_{5}|u_{j}(t_{n})|_{H}|g_{j}(t_{n})|_{V} + |w(t_{n})|_{V^{∗}}|g_{j}(t_{n})|_{V}
≤ M_{5}^{2}ε_{3}|u_{j}(t_{n})|_{H}^{2} + C_{ε_{3}}|g_{j}(t_{n})|_{V}^{2} + \frac{1}{2}|w(t_{n})|_{V^{∗}}^{2} + \frac{1}{2}|g_{j}(t_{n})|_{V}^{2} .

Since |g_{j}(τ)|_{V} ≤ |θ_{j}(τ)|_{V} + |u(τ)|_{V^{∗}} =: M_{6} for all τ ∈ [0, T], we have for any ε_{3} > 0 that

(C_{0} − M_{5}^{2}ε_{3})|u_{j}(t_{n})|_{H}^{2} ≤ ψ^{t_{n}}(w(t_{n})) + |w(t_{n})|_{V^{∗}}^{2} + (C_{ε_{3}} + \frac{1}{2}) |g_{j}(t_{n})|_{V}^{2}
≤ max_{τ∈[0,T]} \{ψ^{τ}(w(τ)) + |w(τ)|_{V^{∗}}^{2}\} + (C_{ε_{3}} + \frac{1}{2}) M_{6}^{2}.

Therefore {u_{j}(t_{n})}_{n=1}^{∞} is bounded in H. Hence, the compact imbedding H ↪ V^{∗} implies that there exist a subsequence {t_{n_{i}}} ⊂ {t_{n}} with t_{n_{i}} → t as i → +∞ and an element z_{j}(t) ∈ V^{∗} such that

u_{j}(t_{n_{i}}) → z_{j}(t) in V^{∗} as i → +∞.  (4.4)

Now, let i → +∞ in the relation

∂ψ^{t_{n_{i}}}(u_{j}(t_{n_{i}})) + F^{−1}u_{j}(t_{n_{i}}) = g_{j}(t_{n_{i}}) in V for all i ∈ N.  (4.5)

Then, (4.4) implies that

∂ψ^{t}(z_{j}(t)) + F^{−1}z_{j}(t) = g_{j}(t) in V.  (4.6)

On the other hand, we have by (4.1) that

g_{j}(t) := θ_{j}(t) + F^{−1}u(t)
= {∂ψ^{t}(u_{j}(t)) + F^{−1}u_{j}(t) − F^{−1}u(t)} + F^{−1}u(t)
= ∂ψ^{t}(u_{j}(t)) + F^{−1}u_{j}(t).  (4.7)

On account of the strictly monotonicity of ∂ψ^{t} + F^{−1}, (4.6) and (4.7) imply that z_{j}(t) = u_{j}(t). Therefore we see that u_{j} is continuous in V^{∗} at t. Thus u_{j} ∈ C([0, T]; V^{∗}). Finally letting j → +∞ in (4.1), we obtain (4.2). □

Lemma 4.2 There exists a subsequence {N_{m}} ⊂ {N_{n}} with N_{m} → +∞ as m → +∞ such that

θ̄^{N_{m}} → θ in L^{2}(0, T; H) as m → +∞.  (4.8)

Proof. Using Lemma 4.1, we consider step functions ū_{j}^{N_{m}} and θ̄_{j}^{N_{m}} given by

ū_{j}^{N_{m}}(t) := u_{j}(t_{k}^{N_{m}}) if t ∈ (t_{k−1}^{N_{m}}, t_{k}^{N_{m}}] for k = 1, 2, …, N_{m},
θ̄_{j}^{N_{m}}(t) := θ_{j}(t_{k}^{N_{m}})
= ∂ψ^{t_{k}^{Nm}}(u_{j}(t_{k}^{N_{m}})) + F^{−1}u_{j}(t_{k}^{N_{m}}) − F^{−1}u(t_{k}^{N_{m}})
if t ∈ (t_{k−1}^{N_{m}}, t_{k}^{N_{m}}] for k = 1, 2, …, N_{m}.

Then, for each j ∈ N we see from the above definitions that

ū_{j}^{N_{m}}(t) → u_{j}(t) in V^{∗} uniformly t ∈ [0, T],  (4.9)
θ̄_{j}^{N_{m}}(t) → θ_{j}(t) in V uniformly t ∈ [0, T],  (4.10)

as m → +∞. Now we note that

I := ∫_{0}^{T} ⟨ū^{N_{m}}(t) − u(t), θ̄^{N_{m}}(t) − θ(t)⟩_{V^{∗},V} dt

= ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ̄^{N_{m}}(t) − θ(t)⟩_{V^{∗},V} dt +∫_{0}^{T} ⟨ū_{j}^{N_{m}}(t) − u_{j}(t), θ̄^{N_{m}}(t) − θ(t)⟩_{V^{∗},V} dt
+ ∫_{0}^{T} ⟨u_{j}(t) − u(t), θ̄^{N_{m}}(t) − θ(t)⟩_{V^{∗},V} dt  (4.11)
=: I_{1} + I_{2} + I_{3}.

The first term I_{1} of the right hand side of (4.11) is estimated as follows: With the help of the strong monotonicity (2.7) in (A1) we have

I_{1} = ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ̄^{N_{m}}(t) − θ(t)⟩_{V^{∗},V} dt

= h_{N_{m}} ∑_{k=1}^{N_{m}}⟨ū^{N_{m}}(t_{k}^{N_{m}}) − ū_{j}^{N_{m}}(t_{k}^{N_{m}}), θ̄^{N_{m}}(t_{k}^{N_{m}}) − θ̄_{j}^{N_{m}}(t_{k}^{N_{m}})⟩_{V^{∗},V}
+ ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ̄_{j}^{N_{m}}(t) − θ_{j}(t)⟩_{V^{∗},V} dt +∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ_{j}(t) − θ(t)⟩_{V^{∗},V} dt
≥ C_{2} ∫_{0}^{T} |θ̄^{N_{m}}(t) − θ̄_{j}^{N_{m}}(t)|_{H}^{2}dt
+ C_{2} ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), F^{−1}u(t) − F^{−1}ū_{j}^{N_{m}}(t)⟩_{V^{∗},V} dt
+ ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ̄_{j}^{N_{m}}(t) − θ_{j}(t)⟩_{V^{∗},V} dt +∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ_{j}(t) − θ(t)⟩_{V^{∗},V} dt
≥ \frac{C_{2}}{3} ∫_{0}^{T} |θ̄^{N_{m}}(t) − θ(t)|_{H}^{2}dt − C_{2} ∫_{0}^{T} |θ̄_{j}^{N_{m}}(t) − θ_{j}(t)|_{H}^{2}dt − C_{2} ∫_{0}^{T} |θ_{j}(t) − θ(t)|_{H}^{2}dt
+ C_{2} ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), F^{−1}u(t) − F^{−1}ū_{j}^{N_{m}}(t)⟩_{V^{∗},V} dt
+ ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ̄_{j}^{N_{m}}(t) − θ_{j}(t)⟩_{V^{∗},V} dt +∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ_{j}(t) − θ(t)⟩_{V^{∗},V} dt.

Adding the second and third terms I_{2} and I_{3} to the above inequality, we have by (4.11) that

\frac{C_{2}}{3} ∫_{0}^{T} |θ̄^{N_{m}}(t) − θ(t)|_{H}^{2}dt

≤ I − I_{2} − I_{3} + C_{2} ∫_{0}^{T} |θ(t) − θ_{j}(t)|_{H}^{2}dt + C_{2} ∫_{0}^{T} |θ_{j}(t) − θ̄_{j}^{N_{m}}(t)|_{H}^{2}dt
− C_{2} ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), F^{−1}u(t) − F^{−1}ū_{j}^{N_{m}}(t)⟩_{V^{∗},V} dt
− ∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ̄_{j}^{N_{m}}(t) − θ_{j}(t)⟩_{V^{∗},V} dt −∫_{0}^{T} ⟨ū^{N_{m}}(t) − ū_{j}^{N_{m}}(t), θ_{j}(t) − θ(t)⟩_{V^{∗},V} dt.

Now, let m → +∞. Then, by convergences (3.16), (3.17), (4.9), (4.10) and the monotonicity of F^{−1},

lim sup_{m→+∞} ∫_{0}^{T} |θ̄^{N_{m}}(t) − θ(t)|_{H}^{2}dt

≤ 3 ∫_{0}^{T} |θ(t) − θ_{j}(t)|_{H}^{2}dt − 3 ∫_{0}^{T} ⟨u(t) − u_{j}(t), F^{−1}u(t) − F^{−1}u_{j}(t)⟩_{V^{∗},V} dt

− \frac{3}{C_{2}} ∫_{0}^{T} ⟨u(t) − u_{j}(t), θ_{j}(t) − θ(t)⟩_{V^{∗},V} dt

≤ 3 ∫_{0}^{T} |θ(t) − θ_{j}(t)|_{H}^{2}dt − \frac{3}{C_{2}} ∫_{0}^{T} ⟨u(t) − u_{j}(t), θ_{j}(t) − θ(t)⟩_{V^{∗},V} dt for all j ∈ N.

(4.12)

Moreover, let j → +∞ in (4.12). Then, by virtue of Lemma 4.1, we obtain that

lim sup_{m→+∞} ∫_{0}^{T} |θ̄^{N_{m}}(t) − θ(t)|_{H}^{2}dt ≤ 0.

Thus θ̄^{N_{m}} → θ in L^{2}(0, T; H) as m → +∞. □

Proof of Theorem 2.1 As was shown above, there exist a subsequence {N_{m}} ⊂ {N}, θ ∈ L^{2}(0, T; V) and u ∈ C([0, T]; V^{∗}) ∩ L^{∞}(0, T; H) such that

θ̄^{N_{m}} → θ in L^{2}(0, T; H),
θ̄^{N_{m}} → θ weakly in L^{2}(0, T; V),
u^{N_{m}} → u in C([0, T]; V^{∗}) as m → +∞,

and

θ(t) = ∂ψ^{t}(u(t)) in V for a.e. t ∈ [0, T].

Now, define V^{∗}-valued step functions K̄^{N_{m}}, Ḡ^{N_{m}} and f̄^{N_{m}} on [0, T] by

K̄^{N_{m}}(t) = K(t_{k}^{N_{m}}, θ_{k}^{N_{m}}) if t ∈ (t_{k−1}^{N_{m}}, t_{k}] for k = 1, 2, …, N_{m},
Ḡ^{N_{m}}(t) = G(t_{k}^{N_{m}}, u_{k}^{N_{m}}) if t ∈ (t_{k−1}^{N_{m}}, t_{k}] for k = 1, 2, …, N_{m},
f̄^{N_{m}}(t) = f_{k}^{N_{m}} if t ∈ (t_{k−1}^{N_{m}}, t_{k}] for k = 1, 2, …, N_{m}.

Then our approximate problem (DN)_{N_{m}} can be written in the form:

\frac{d}{dt}u^{N_{m}}(t) + K̄^{N_{m}}(t) + Ḡ^{N_{m}}(t) = f̄^{N_{m}}(t) in V^{∗} for a.e. t ∈ [0, T].  (4.13)

Our claim is to show that (4.13) converges to (2.1) as m → +∞. To this end, we show that

K̄^{N_{m}} → K(·, θ(·)) weakly in L^{2}(0, T; V^{∗}) as m → +∞.  (4.14)

By (2.10) in the assumptions (A2) of K we see that

K(·, θ̄^{N_{m}}(·)) → K(·, θ(·)) weakly in L^{2}(0, T; V^{∗}) as m → +∞.

Moreover, by (2.11) in the assumptions (A2) of K,

|K̄^{N_{m}}(t) − K(t, θ̄^{N_{m}}(t))|_{V^{∗}} ≤ ω(|t_{k}^{N_{m}} − t|)(|θ̄^{N_{m}}(t)|_{V} + 1)

for all t ∈ (t_{k−1}^{N_{m}}, t_{k}^{N_{m}}] and k = 1, 2, …, N_{m}.

Hence (4.14) holds. Similarly, by (2.13) in the assumption (A3) of G, we have

Ḡ^{N_{m}} → G(·, u(·)) weakly in L^{2}(0, T; V^{∗}) as m → +∞,  (4.15)

and clearly

f̄^{N_{m}} → f in L^{2}(0, T; V^{∗}) as m → +∞.  (4.16)

These convergences imply that {(d/dt)u^{N_{m}}} is bounded in L^{2}(0, T; V^{∗}), so that

\frac{d}{dt}u^{N_{m}} → u′ weakly in L^{2}(0, T; V^{∗}) as m → +∞,

Finally, passing to the limit in m of (4.13), we have

u′(t) + K(t, θ(t)) + G(t, u(t)) = f(t) in V^{∗} for a.e. t ∈ [0, T].

Thus {u, θ} is a solution of (DN). □

5 Application

In this section we give an application of our abstract result.

Let 0 < T < +∞, and Ω ⊂ R^{3} be a bounded domain with smooth boundary Γ := ∂Ω. Assume that for each t ∈ [0, T], Ω is divided into two subdomains Ω_{1}(t) and Ω_{2}(t) by a time-dependent interface Γ_{12}(t), that is,

Ω = Ω_{1}(t) ∪ Γ_{12}(t) ∪ Ω_{2}(t) for all t ∈ [0, T].

We denote by Γ_{i}(t) the set ∂Ω_{i}(t) \ Γ_{12}(t) for i = 1, 2 (see Fig. 1).

[figure: Fig.1; ν^{−}; ν^{+}; ν; Ω_{2}(t); Γ_{12}(t); Ω_{1}(t); S(t); Γ_{2}(t); Γ_{1}(t)]

Under these setting of the domains, we consider the following system of two degenerate parabolic equations:

(\frac{∂u_{1}}{∂t} + v_{1} · ∇u_{1}) − div(k_{1}(θ_{1})∇θ_{1}) = h in Q_{1} := ⋃_{t∈(0,T)}\{t\} × Ω_{1}(t),  (5.1)
θ_{1} = β_{1}(u_{1}) in Q_{1},  (5.2)
(\frac{∂u_{2}}{∂t} + v_{2} · ∇u_{2}) − div(k_{2}(θ_{2})∇θ_{2}) = h in Q_{2} := ⋃_{t∈(0,T)}\{t\} × Ω_{2}(t),  (5.3)
θ_{2} = β_{2}(u_{2}) in Q_{2},  (5.4)

where div means the divergence with respect to space variable x, h is a given function in Q := (0, T) × Ω, and v_{1} and v_{2} are given vector fields on Q. Moreover, for each i = 1, 2, β_{i} and k_{i} are functions such that

• β_{i} : R → R is Lipschitz continuous and non-decreasing, and satisfies that

|β_{i}(r)| ≥ C_{β_{i}}|r| − C′_{β_{i}} for all r ∈ R,  (5.5)

where C_{β_{i}} and C′_{β_{i}} are positive constants;

• k_{i} : R → R is positive and non-decreasing, and satisfies that

0 < C_{k_{i}} ≤ k_{i}(r) ≤ C′_{k_{i}} for all r ∈ R,  (5.6)

where C_{k_{i}} and C′_{k_{i}} are positive constants.

We combine these two equations (5.1) and (5.3) on the moving boundary Γ_{12}(t) by the transmission condition

θ_{1} = θ_{2}, k_{1}(θ_{1})\frac{∂θ_{1}}{∂ν^{+}} + k_{2}(θ_{2})\frac{∂θ_{2}}{∂ν^{−}} = 0 on Σ_{12} := ⋃_{t∈(0,T)}\{t\} × Γ_{12}(t),  (5.7)

where ν^{+} := ν^{+}(t, x) is the unit normal vector on Γ_{12}(t) pointing to Ω_{2}(t) and ν^{−} := −ν^{+}. We consider the system {(5.1)-(5.7)}, subject to the initial and boundary conditions for i = 1, 2:

u_{i}(0) = u_{i0} in Ω_{i}(0),  (5.8)
k_{i}(θ_{i})\frac{∂θ_{i}}{∂ν} + n_{0}θ_{i} = p on Σ_{i} := ⋃_{t∈(0,T)}\{t\} × Γ_{i}(t),  (5.9)

where u_{0} and p are given functions, n_{0} is a positive constant and ν := ν(x) is the unit outward normal vector on Γ. The system {(5.1)-(5.9)} is referred as (TDP).

It is well-known that equations (5.1) and (5.2) (resp. (5.3) and (5.4)) describe a Stefan problem in non-cylindrical domain Q_{1} (resp. Q_{2}), in which there is a convective vector field. We call such problems the transmission-Stefan problem. In Fukao, Kenmochi and Pawłow [10], [11], this problem was discussed under the following assumptions (A4) and (A5):

(A4) The domain Ω_{1}(t) and Ω_{2}(t) depend smoothly on time t in the following sense: There is a transformation y = y(t, x) := (y_{1}(t, x), y_{2}(t, x), y_{3}(t, x)) which is a function of C^{2}-class from \overline{Q} into R^{3} satisfying that y = y(t, x) is a C^{2}-diffeomorphism from \overline{Ω} onto itself for each t ∈ [0, T] and

y(t, \overline{Ω_{i}(t)}) = \overline{Ω_{i}(0)} for all t ∈ [0, T] for i = 1, 2, y(0, ·) = I (identity) in \overline{Ω}.

(A5) The convective vector fields v_{1} and v_{2}, which may occur by the motion of domains, are prescribed so that v_{i} ∈ C^{1}(\overline{Q_{i}})^{3} for i = 1, 2 and the following properties are satisfied:

divv_{i}(t, ·) = 0 in Ω_{i}(t) for all t ∈ (0, T) for i = 1, 2,  (5.10)
v_{1} · ν^{+} = −v_{2} · ν^{−} = v_{Σ_{12}} on Σ_{12},  (5.11)
v_{1} · ν = v_{2} · ν = 0 on Σ := (0, T) × Γ,  (5.12)

where v_{Σ_{12}} := v_{Σ_{12}}(t, x) is the normal speed of Γ_{12}(t) at x ∈ Γ_{12}(t).

Using our abstract theory, we show the existence of a weak solution for (TDP). Let V be the Sobolev space H^{1}(Ω) and H be the Hilbert space L^{2}(Ω). Then we have

H^{1}(Ω) ↪^{d} L^{2}(Ω) ↪^{d}(H^{1}(Ω))^{∗},

where (H^{1}(Ω))^{∗} is the dual space of H^{1}(Ω). Moreover ψ^{t}, K, G and f are defined as follows:

• ψ^{t} is a proper, lower semicontinuous and convex function on V^{∗} defined by

ψ^{t}(u) := \{ \begin{matrix}∑_{i=1,2} ∫_{Ω_{i}(t)} β̂_{i}(u)dx & if u ∈ H, \\ +∞ & if u ∈ V^{∗} \ H, \end{matrix}

where β̂_{i}(r) := ∫_{0}^{r} β_{i}(s)ds;

• K is a singlevalued operator from [0, T] × V into V^{∗} defined by

⟨K(t, θ), z⟩_{V^{∗},V} := ∫_{Ω} k(t, θ)(∇θ · ∇z)dx + n_{0} ∫_{Γ} θzdΓ for all z ∈ V,

where k(t, θ) := k(t, x, θ(x)) and k(t, x, ·) : R → R is defined by

k(t, x, r) := \{ \begin{matrix}k_{1}(r) for all r ∈ R & if x ∈ Ω_{1}(t), \\ k_{2}(r) for all r ∈ R & if x ∈ \overline{Ω_{2}(t)}; \end{matrix}

• G is a singlevalued operator from [0, T] × H into V^{∗} defined by

⟨G(t, u), z⟩_{V^{∗},V} := − ∑_{i=1,2} ∫_{Ω_{i}(t)} u(v_{i}(t) · ∇z)dx for all z ∈ V;

• f is a function in L^{2}(0, T; V^{∗}) given by

⟨f(t), z⟩_{V^{∗},V} := ∫_{Ω} h(t)zdx + ∫_{Γ} p(t)zdΓ for all z ∈ V and a.e. t ∈ [0, T],

where h ∈ L^{2}(Q) and p ∈ L^{2}(Σ) are prescribed.

It is easily seen that (A1), (A2) and (A3) are satisfied. Now our problem (TDP) is reformulated in the following weak variational form (cf. Fukao, Kenmochi and Pawłow [10]):

− ∫_{Q} u\frac{∂η}{∂t} dxdt − ∫_{Q} u(v · ∇η)dxdt + ∫_{Q} k(θ)(∇θ · ∇η)dxdt + n_{0} ∫_{Σ} θηdΓdt

= ∫_{Q} hηdxdt + ∫_{Σ} pηdΓdt + ∫_{Ω} u_{0}η(0)dx for all η ∈ W,  (5.13)

where

k(θ) := k(t, x, θ(t, x)),
W := {η ∈ H^{1}(Q); η(T, ·) = 0 on Ω},

and

u := u_{i}, θ := θ_{i}, v := v_{i} on Q_{i} and u_{0} := u_{i0} on Ω_{i}(0) for i = 1, 2.

In fact, the variational identity (5.13) is derived from (5.1), (5.2), (5.3) and (5.4) in the following way. Let η be any test function in W. First, multiply (5.1) and (5.3) by η and integrate their resultants over Q_{1} and Q_{2}, respectively. Then, with the help of conditions (5.8), (5.10), (5.11), (5.12) and the Green-Stokes’ formula we have

∑_{i=1,2} ∫_{Q_{i}} (\frac{∂u_{i}}{∂t} + v_{i} · ∇u_{i}) ηdxdt

= − ∫_{Q_{1}} u_{1} (\frac{∂η}{∂t} + v_{1} · ∇η) dxdt + ∫_{Σ_{12}} u_{1}η(v_{1} · ν^{+})dΓ_{12}(t)dt − ∫_{Ω_{1}(0)} u_{10}η(0)dx

− ∫_{Q_{2}} u_{2} (\frac{∂η}{∂t} + v_{2} · ∇η) dxdt + ∫_{Σ_{12}} u_{2}η(v_{2} · ν^{−})dΓ_{12}(t)dt − ∫_{Ω_{2}(0)} u_{20}η(0)dx

= − ∫_{Q} u (\frac{∂η}{∂t} + v · ∇η) dxdt − ∫_{Ω} u_{0}η(0)dx.  (5.14)

Next, by (5.7) and (5.9),

− ∑_{i=1,2} ∫_{Q_{i}} div(k_{i}(θ_{i})∇θ_{i})ηdxdt

= ∫_{Q_{1}} k_{1}(θ_{1})(∇θ_{1} · ∇η)dxdt − ∫_{Σ_{1}} k_{1}(θ_{1})\frac{∂θ_{1}}{∂ν} ηdΓ_{1}(t)dt + ∫_{Σ_{12}} k_{1}(θ_{1})\frac{∂θ_{1}}{∂ν^{+}} ηdΓ_{12}(t)dt

+ ∫_{Q_{2}} k_{2}(θ_{2})(∇θ_{2} · ∇η)dxdt − ∫_{Σ_{2}} k_{2}(θ_{2})\frac{∂θ_{2}}{∂ν} ηdΓ_{2}(t)dt + ∫_{Σ_{12}} k_{2}(θ_{2})\frac{∂θ_{2}}{∂ν^{−}} ηdΓ_{12}(t)dt

= ∫_{Q} k(θ)(∇θ · ∇η)dxdt + ∫_{Σ}(n_{0}θ − p)ηdΓ(t)dt.  (5.15)

Here, add (5.14) and (5.15). Then we get the variational identity (5.13). Moreover by (5.2), (5.4) and the definition of ψ^{t} we see that

θ(t) = ∂ψ^{t}(u(t)) in V for a.e. t ∈ [0, T],

and (5.13) can be transformed to the doubly nonlinear problem (DN) in the space V^{∗}. Therefore, applying Theorem 2.1, we conclude that there exist u ∈ W^{1,2}(0, T; V^{∗}) ∩ L^{∞}(0, T; H) and θ ∈ L^{2}(0, T; V) which give a weak solution of (TDP) in the variational sense (5.13).

References

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Masayasu Aso

Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan E-mail: aso@math.e.chiba-u.ac.jp

Takesi Fukao

Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan E-mail: t-fukao@math.e.chiba-u.ac.jp

Nobuyuki Kenmochi

Department of Mathematics, Faculty of Education, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan E-mail: kenmochi@math.e.chiba-u.ac.jp

Technical Reports of Mathematical Sciences

Chiba University, vol. 18 (2002)

No. 1 Convergence of approximate solutions in the sense of distribution, by H. Koshigoe and K. Kitahara.

No. 2 Growth of transcendental entire solution of some q-difference equation, by I. Eli, K. Gotoh and N. Yanagihara.

No. 3 Stability analysis for Allen-Cahn type equation associated with the total variation energy, by K. Shirakawa and M. Kimura.

No. 4 Hilbert coefficients and Buchsbaumness of associated graded rings, by S. Goto and K. Nishida.

No. 5 On the extended Haagerup tensor product in operator spaces, by T. Itoh and M. Nagisa.

No. 6 Topological spectrum of locally compact Cantor minimal systems, by H. Matui.

No. 7 Some remarks on topological orbit equivalence of Cantor minimal systems, by H. Matui.

No. 8 Simultaneous estimation of scale matrices in two-sample problem under elliptically contoured distributions, by H. Tsukuma and Y. Konno.

No. 9 Entire solutions of q-difference equations, by I. Eli and N. Yanagihara.

No. 10 Meromorphic solutions of q-difference equations, by I. Eli and N. Yanagihara.

No. 11 On estimation problem in multivariate linear calibration with elliptical errors, by H. Tsukuma.

No. 12 Alternative estimators of the common regression matrix in two GMANOVA models under weighted quadratic losses, by H. Tukuma and Y. Konno.

No. 13 A new class of doubly nonlinear evolution equations, by M. Aso, T. Fukao and N. Kenmochi.

Correspondences concerning these Technical Reports should be addressed to

Koji Nishida Graduate School of Science and Technology Chiba University Yayoi-cho 1-33, Inage-ku, Chiba-shi, 263-8522 JAPAN. e-mail: nishida@math.s.chiba-u.ac.jp