Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 22 1-11; http://www.math.u-szeged.hu/ejqtde/

On contact problems for nonlinear parabolic functional differential equations

László Simon^{∗}

Abstract The results of [3] by W. Jäger and N. Kutev on a nonlinear elliptic transmission problem are extended (in a modified way) to nonlinear parabolic problems with nonlinear and nonlocal contact conditions.

Introduction

In [3] W. Jäger and N. Kutev considered the following nonlinear transmission (contact) problem for nonlinear elliptic equations:

∑_{i=1}^{n} D_{i}[a_{i}(x, u, Du)] + b(x, u, Du) = 0 in Ω  (0.1)

u = g on ∂Ω  (0.2)
[∑_{i=1}^{n} a_{i}(x, u, Du)ν_{i}]|_{S} = 0  (0.3)
u_{1} = Φ(u_{2}) on S  (0.4)

where Ω ⊂ R^{n} is a bounded domain with sufficiently smooth boundary ∂Ω which is divided into two subdomains Ω_{1}, Ω_{2} by means of a smooth surface S which has no intersection point with ∂Ω, the boundary of Ω_{1} is S and the boundary of Ω_{2} is S ∪∂Ω. Further, [f]|_{S} denotes the jump of f on S in the direction of the normal ν, Φ is a smooth strictly increasing function and u_{j} denotes the restriction of u to Ω_{j} (j = 1, 2). The coefficients of the equation are smooth in \overline{Ω_{j}} and satisfy standard conditions but they have jump on the surface S. The problem was motivated e.g. by reaction-diffusion phenomena in porous medium. The authors formulated conditions which implied comparison principles, existence and uniqueness of the weak and the classical solution, respectively.

The aim of this paper is to consider nonlinear parabolic functional differential equations with a modified contact condition on S: with boundary conditions of third type, containing delay. In [7] we studied parabolic differential equations with contact conditions, considered in [3]. In Section 1 we shall prove existence and uniqueness theorems and in Section 2 we shall formulate a theorem on boundedness of the solutions and a stabilization result.

∗This work was supported by the Hungarian National Foundation for Scienific Research under grant OTKA T 031807. This paper is in final form and no version of it will be submitted for publication elsewhere.

1 Existence and uniqueness of solutions

Let Ω ⊂ R^{n} be a bounded domain having the uniform C^{1} regularity property (see [1]) which is divided into two subdomains Ω_{1}, Ω_{2} by means of a smooth surface S which has no intersection point with ∂Ω, the boundary of Ω_{1} is S and the boundary of Ω_{2} is S ∪ ∂Ω (such that Ω_{1} and Ω_{2} have the C^{1} regularity property).

We shall consider weak solutions of the problem

D_{t}u^{j} −∑_{i=1}^{n} D_{i}[a_{i}^{j}(t, x, u^{j}, Du^{j})]+b^{j}(t, x, u^{j}, Du^{j})+G^{j}(u^{1}, u^{2}) = F^{j}(t, x), (1.5)

(t, x) ∈ Q_{T}^{j} = (0, T) × Ω_{j}, j = 1, 2
u^{2} = 0 on Γ_{T} = [0, T] × ∂Ω  (1.6)
∑_{i=1}^{n} a_{i}^{j}(t, x, u^{j}, Du^{j})ν_{i}^{j}|_{S_{T}} = H^{j}(u^{1}, u^{2}), S_{T} = [0, T] × S, j = 1, 2 (1.7)
u(0, x) = 0, x ∈ Ω_{1} ∪ Ω_{2}  (1.8)

where u^{j} = u|_{Q_{T}^{j}}, G^{j}, H^{j} are operators (which will be defined below as well as

functions F^{1}, F^{2} ), ν^{j} = (ν_{1}^{j}, ..., ν_{n}^{j}) is the normal unite vector on S (ν^{1} = −ν^{2}), a_{i}^{j}, b^{j} have certain polynomial growth in u^{j}, Du^{j}.

Let p ≥ 2 be a real number. For any domain Ω_{0} ⊂ R^{n} denote by W^{1,p}(Ω_{0}) the usual Sobolev space of real valued functions with the norm

∥ u ∥= [∫_{Ω_{0}}(|Du|^{p} + |u|^{p})]^{1/p} .

Let V_{1} = W^{1,p}(Ω_{1}), V_{2} = {w ∈ W^{1,p}(Ω_{2}) : w|_{∂Ω} = 0} and V = V_{1} × V_{2}. Denote by L^{p}(0, T; V) the Banach space of the set of measurable functions u = (u^{1}, u^{2}) : (0, T) → V such that ∥ u ∥^{p} is integrable and define the norm by

∥ u ∥_{L^{p}(0,T;V)}^{p}= ∫_{0}^{T} ∥ u(t) ∥_{V}^{p} dt.

The dual space of L^{p}(0, T; V) is L^{q}(0, T; V^{⋆}) where 1/p + 1/q = 1 and V^{⋆} is the dual space of V (see, e.g., [4], [8]).

Now we formulate the conditions with respect to the problem (1.5) - (1.8) and the existence theorem on the weak solutions of this problem where F = (F^{1}, F^{2}) ∈ L^{q}(0, T; V^{⋆}).

Assume that I. The functions a_{i}^{j}, b^{j} : Q_{T}^{j} ×R^{n+1} → R satisfy the Carathéodory conditions,

i.e. a_{i}^{j}(t, x, η, ζ), b^{j}(t, x, η, ζ) are measurable in (t, x) ∈ Q_{T}^{j} = (0, T)×Ω_{j} for each fixed (η, ζ) ∈ R^{n+1} and they are continuous in (η, ζ) ∈ R^{n+1} for a.e. (t, x) ∈ Q_{T}^{j}.

II. |a_{i}^{j}(t, x, η, ζ)| ≤ c_{1}[|η|^{p−1} + |ζ|^{p−1}] + k_{1}^{j}(x), for a.e. (t, x) ∈ Q_{T}^{j}, each (η, ζ) ∈ R^{n+1} with some constant c_{1} and a function k_{1}^{j} ∈ L^{q}(Ω_{j}),

|b^{j}(t, x, η, ζ)| ≤ c_{1}[|η|^{p−1} + |ζ|^{p−1}] + k_{1}^{j}(x). III. ∑_{i=1}^{n}[a_{i}^{1}(t, x, η, ζ) − a_{i}^{1}(t, x, η, ζ^{⋆})](ζ_{i} − ζ_{i}^{⋆}) > 0 if ζ ≠ ζ^{⋆}. IV. ∑_{i=1}^{n} a_{i}^{j}(t, x, η, ζ)ζ_{i} + b^{j}(t, x, η, ζ)η ≥ c_{2}[|ζ|^{p} + |η|^{p}] − k_{2}^{1}(x), (t, x) ∈ Q_{T}^{1} ∑_{i=1}^{n} a_{i}^{2}(t, x, η, ζ)ζ_{i} + b^{2}(t, x, η, ζ)η ≥ c_{2}|ζ|^{p} − k_{2}^{2}(x), (t, x) ∈ Q_{T}^{2} with some constant c_{2} > 0, k_{2}^{j} ∈ L^{1}(Ω_{j}). V. G^{j} : L^{p}(Q_{T}^{1} ) × L^{p}(Q_{T}^{2} ) → L^{q}(Q_{T}^{j}) are bounded (nonlinear) operators

which are demicontinuous (i.e. (u_{k}) → u with respect to the norm L^{p}(Q_{T}^{1} ) × L^{p}(Q_{T}^{2} ) implies that G^{j}(u_{k}) → G^{j}(u) weakly in L^{q}(Q_{T}^{j}).

VI. H^{j} : L^{p}(0, T; V) → L^{q}(S_{T}) are bounded (nonlinear) operators having the following property: There exists a positive number δ < 1 − 1/p such that the operators H^{j} are demicontinuous from L^{p}(0, T; W^{1−δ,p}(Ω_{1}) × W^{1−δ,p}(Ω_{2})) into L^{q}(S_{T}).

VII. lim_{∥u∥→∞} ^{\frac{∥G^{j}(u)∥_{L^{q} (Q{j \atop T} )}^{q}+∥H^{j}(u)∥_{L^{q} (S_{T} )}^{q}}{∥u∥_{L^{p}(0,T;V )}^{p}}} = 0

for any u ∈ L^{p}(0, T; V). Then we may define the operators A^{j} : L^{p}(0, T; V_{j}) → L^{q}(0, T; V_{j}^{⋆}) by

[A^{j}(u^{j}), v^{j}] = ∫_{Q_{T}^{j}} [∑_{i=1}^{n} a_{i}^{j}(t, x, u^{j}, Du^{j})D_{i}v^{j} + b^{j}(t, x, u^{j}, Du^{j})v^{j}] dtdx,

= ∫_{0}^{T} ⟨A^{j}(u^{j})(t), v^{j}(t)⟩dt,
A = (A^{1}, A^{2}) : L^{p}(0, T; V) → L^{q}(0, T; V^{⋆})
by [A(u), v] = [A^{1}(u^{1}), v^{1}] + [A^{2}(u^{2}), v^{2}]

and the operators B^{j} : L^{p}(0, T; V) → L^{q}(0, T; V_{j}^{⋆}) by

[B^{j}(u), v^{j}] = [B_{1}^{j}(u), v^{j}] − [B_{2}^{j}(u), v^{j}] =
∫_{Q_{T}^{j}} G^{j}(u)v^{j}dtdx − ∫_{S_{T}} H^{j}(u)v^{j}dtdσ_{x},
u = (u^{1}, u^{2}) ∈ L^{p}(0, T; V), (v^{1}, v^{2}) ∈ L^{p}(0, T; V).

By I, II, V, VI, Hölder’s inequality and Vitali’s theorem operator

A + B = (A^{1}, A^{2}) + (B^{1}, B^{2}) : L^{p}(0, T; V) → L^{q}(0, T; V^{⋆})

is bounded (i.e. it maps bounded sets of L^{p}(0, T; V) into bounded sets of L^{q}(0, T; V^{⋆})) and demicontinuous.

Theorem 1.1 Assume I - VII. Then for any F = (F^{1}, F^{2}) ∈ L^{q}(0, T; V^{⋆}) there exists u = (u^{1}, u^{2}) ∈ L^{p}(0, T; V) such that D_{t}u^{j} ∈ L^{q}(0, T; V_{j}^{⋆}),

D_{t}u^{j} + A^{j}(u^{j}) + B^{j}(u^{1}, u^{2}) = F^{j}, j = 1, 2  (1.9)
u^{j}(0) = 0, j = 1, 2.  (1.10)

Remark 1 If u satisfies (1.9), (1.10), we say that u = (u^{1}, u^{2}) is a weak solution of (1.5) - (1.8).

Proof of Theorem 1.1 Let the operators L^{j} : L^{p}(0, T; V_{j}) → L^{q}(0, T; V_{j}^{⋆}) be defined by

D(L^{j}) = {u^{j} ∈ L^{p}(0, T; V_{j}) : D_{t}u^{j} ∈ L^{q}(0, T; V_{j}^{⋆}), u^{j}(0) = 0},
[L^{j}u^{j}, v^{j}] = ∫_{0}^{T} ⟨D_{t}u^{j}(t, ·).v^{j}(t, ·)⟩dt, u^{j} ∈ D(L^{j}), v^{j} ∈ L^{p}(0, T; V_{j})

where D_{t}u^{j} is the distributional derivative of u^{j}. It is well known that L^{j} is a closed linear maximal monotone map (see, e.g., [8]), thus L = (L^{1}, L^{2}) : L^{p}(0, T; V) → L^{q}(0, T; V^{⋆}) is a closed linear maximal monotone map, too. Therefore, Theorem 1. will follow from Theorem 4. of [2] if we show that operator A + B is coercive and pseudomonotone with respect to D(L). It is known that A is pseudomonotone with respect to D(L) (see, e.g. [5]). The latter property means that for any sequence (u_{k}) in D(L) with

(u_{k}) → u weakly in L^{p}(0, T; V),  (1.11)
(Lu_{k}) → Lu weakly in L^{q}(0, T; V^{⋆}),  (1.12)
lim sup_{k→∞}[A(u_{k}), u_{k} − u] ≤ 0  (1.13)

we have

lim_{k→∞}[A(u_{k}), u_{k} − u] = 0,  (1.14)
(A(u_{k})) → A(u) weakly in L^{q}(0, T; V^{⋆}).  (1.15)

Now we prove that (A + B) is pseudomonotone with respect to D(L), too. Assume (1.11), (1.12) and

lim sup_{k→∞}[(A + B)(u_{k}), u_{k} − u] ≤ 0.  (1.16)

Since the imbedding W^{1,p}(Ω_{j}) → W^{1−δ,p}(Ω_{j}) is compact, by a well known compactness result (see, e.g., [4]) (1.11), (1.12) imply that there is a subsequence (u_{k_{l}}) of (u_{k}) such that

(u_{k_{l}}) → u in L^{p}(0, T; W^{1−δ,p}(Ω_{1}) × W^{1−δ,p}(Ω_{2})).  (1.17)

Since the trace operators W^{1−δ,p}(Ω_{j}) → L^{p}(S) are continuous (δ +1/p < 1, see, e.g., [1]), we obtain by (1.17), V, VI and Hölder’s inequality

lim_{l→∞}[B(u_{k_{l}}), u_{k_{l}} − u] = 0.  (1.18)

Further, (1.17), V, VI imply

(B(u_{k_{l}})) → A(u) weakly in L^{q}(0, T; V^{⋆}).  (1.19)

From (1.16), (1.18) we obtain

lim sup_{l→∞}[A(u_{k_{l}}), u_{k_{l}} − u] ≤ 0.  (1.20)

As A is pseudomonotone with respect to D(L), (1.11), (1.12) and (1.20) imply

lim_{l→∞}[A(u_{k_{l}}), u_{k_{l}} − u] = 0,  (1.21)
(A(u_{k_{l}})) → A(u) weakly in L^{q}(0, T; V^{⋆}).  (1.22)

Finally, from (1.18), (1.19), (1.21) and (1.22) we obtain

lim_{l→∞}[(A + B)(u_{k_{l}}), u_{k_{l}} − u] = 0,  (1.23)
((A + B)(u_{k_{l}})) → (A + B)(u) weakly in L^{q}(0, T; V^{⋆})  (1.24)

which means that (A + B) is pseudomonotone with respect to D(L). (It is easy to show that (1.23), (1.24) hold for the sequence (u_{k}), too.)

Now we show that A + B is coercive. By assumption IV we have

[A(u), u] ≥ c_{2} ∥ u ∥_{L^{p}(0,T;V)}^{p} −c_{2}^{⋆}  (1.25)

with constants c_{2} > 0, c_{2}^{⋆}. Further, assumption VII implies

\frac{|[B(u), u]|}{∥ u ∥^{p}} ≤ \frac{∥ B(u) ∥}{∥ u ∥^{p−1}} = [\frac{∥ B(u) ∥^{q}}{∥ u ∥^{p}} ]^{1/q} → 0  (1.26)

if ∥ u ∥→ 0. Thus by (1.25), (1.26)

\frac{[(A + B)(u), u]}{∥ u ∥} ≥ \frac{[A(u), u]}{∥ u ∥} − \frac{|[B(u), u]|}{∥ u ∥} ≥

\frac{c_{2} ∥ u ∥^{p} −c_{2}^{⋆}}{∥ u ∥} − \frac{|[B(u), u]|}{∥ u ∥^{p}} ∥ u ∥^{p−1}=

∥ u ∥^{p−1} [c_{2} − \frac{|[B(u), u]|}{∥ u ∥^{p}}] − \frac{c_{2}^{⋆}}{∥ u ∥} → +∞

if ∥ u ∥→ ∞, i.e. A + B is coercive.

Examples for G^{j} and H^{j}

a/ Let

[G^{1}(u)](t, x) = γ^{1}(t, x, u^{1}(χ_{1}(t), x), ∫_{Ω_{2}} d^{2}(y)u^{2}(χ_{2}(t), y)dy), (t, x) ∈ Q_{T}^{1} ,

[G^{2}(u)](t, x) = γ^{2}(t, x, ∫_{Ω_{1}} d^{1}(y)u^{1}(χ_{1}(t), y)dy, u^{2}(χ_{2}(t), x)), (t, x) ∈ Q_{T}^{2} where χ_{1}, χ_{2} are C^{1} functions satisfying χ′_{j} > 0, 0 ≤ χ_{j}(t) ≤ t; d^{1}, d^{2} are L^{∞} functions; the functions γ^{j} satisfy the Carathéodory conditions and

|γ_{j}(t, x, θ_{1}, θ_{2})| ≤ c^{j}(θ_{1}, θ_{2})|θ|^{p−1} + k_{1}^{j}(x)  (1.27)

with continuous functions c^{j} having the property

lim_{|(θ_{1},θ_{2})|→∞} c^{j} = 0, k_{1}^{j} ∈ L^{q}(Ω_{j}).

By using Hölder’s inequality and Vitali’s theorem it is not difficult to prove that condition V is fulfilled (see [5], [6]) and by (1.27) one obtains VII.

b/ Similarly can be considered operators

[G^{1}(u)](t, x) = ∫_{0}^{t} γ^{1} (t, τ, x, u^{1}(τ, x), ∫_{Ω_{2}} d^{2}(y)u^{2}(τ), y)dy) dτ, (t, x) ∈ Q_{T}^{1} ,

[G^{2}(u)](t, x) = ∫_{0}^{t} γ^{2} (t, τ, x, ∫_{Ω_{1}} d^{1}(y)u^{1}(τ, y)dy, u^{2}(τ, x)) dτ, (t, x) ∈ Q_{T}^{2}

where γ^{j} satisfy a condition which is analogous to (1.27).

c/ Let

[H^{j}(u)](t, x) = h^{j}(t, x, u^{1}(χ_{1}(t), x), u^{2}(χ_{2}(t), x)), (t, x) ∈ S_{T} ,

where the functions h^{j} satisfy a condition analogous to (1.27). By δ < 1 − 1/p the trace operator W^{1−δ,p}(Ω) → L^{p}(∂Ω) is bounded, thus by using Hölder’s inequality and Vitali’s theorem, one can prove that VI and by the condition, analogous to (1.27), VII are satisfied.

Similarly can be treated the following examples:

d/

[H^{j}(u)](t, x) = ∫_{0}^{t} h^{j}(t, τ, x, u^{1}(τ, Φ_{1}(x)), u^{2}(τ, Φ_{2}(x)))dτ, (t, x) ∈ S_{T} ,

where Φ_{j}, (Φ_{j})^{−1} are C^{1} functions in a neighbourhood of S, Φ_{j}(S) = S. e/

[H^{j}(u)](t, x) = h^{j}(t, x, ∫_{S} u^{1}(χ(t), y)dσ_{y}, ∫_{S} u^{2}(χ(t), y)dσ_{y}), (t, x) ∈ S_{T} ,

f/

[H^{j}(u)](t, x) = ∫_{0}^{t} h^{j} (t, τ, x, ∫_{S} u^{1}(τ, y)dσ_{y}, ∫_{S} u^{2}(τ, y)dσ_{y}) dτ, (t, x) ∈ S_{T} .

By using monotonicity arguments one can prove uniqueness of the solution.

Theorem 1.2 Assume that

∑_{i=1}^{n}[a_{i}^{j}(t, x, η, ζ) − a_{i}^{j}(t, x, η^{⋆}, ζ^{⋆})](ζ_{i} − ζ_{i}^{⋆})+  (1.28)
[b^{j}(t, x, η, ζ) − b^{j}(t, x, η^{⋆}, ζ^{⋆})](η − η^{⋆}) ≥ −c_{0}(η − η^{⋆})^{2}

with some constant c_{0} and

∑_{j=1}^{n}[H^{j}(u) − H^{j}(v), u − v] ≥ 0, u, v ∈ L^{p}(0, T; V).  (1.29)

Further, for the operators

[G̃^{j}(ũ)](t, x) = e^{−αt}[G^{j}(e^{αt}ũ)](t, x),

the inequality

∥ G̃^{j}(ũ) − G̃^{j}(ṽ) ∥_{L^{2}(Q_{T}^{j})}≤ c̃ ∥ ũ − ṽ ∥_{L^{2}(Q_{T}^{1} )×L^{2}(Q_{T}^{2} )}  (1.30)

holds where the constant c̃ is not depending on the positive number α and ũ, ṽ. Then the problem (1.9), (1.10) may have at most one solution.

Remark 2 It is easy to show that (1.30) holds for the above examples a/ and b/ if functions γ^{j} satisfy (global) Lipschitz condition with respect to θ_{1} and θ_{2}. Further, (1.29) holds if [H^{j}(u)](t, x) = h^{j}(t, x, u^{1}(t, x), u^{2}(t, x)) and

∑_{j=1}^{2}[h^{j}(t, x, θ_{1}, θ_{2}) − h^{j}(t, x, θ_{1}^{⋆}, θ_{2}^{⋆})](θ_{j} − θ_{j}^{⋆}) ≥ 0.

(E.g. h^{1} is not depending on θ_{2}, h^{2} is not depending on θ_{1} and for a.e. fixed (t, x) the functions θ_{j} → h^{j}(t, x, θ_{j}) are monotone increasing.)

The proof of Theorem 1.2 Perform the substitution u = e^{αt}ũ. Then (1.9), (1.10) is equivalent with

D_{t}ũ^{j} + Ã^{j}(ũ^{j}) + B̃^{j}(ũ^{1}, ũ^{2}) + αũ^{j} = F̃^{j},  (1.31)
ũ^{j}(0) = 0.  (1.32)

where

[Ã^{j}(ũ^{j}), v^{j}] = ∫_{Q_{T}^{j}} [∑_{i=1}^{n} ã_{i}^{j}(t, x, ũ^{j}, Dũ^{j})D_{i}v^{j} + b̃^{j}(t, x, ũ^{j}, Dũ^{j})v^{j}] dtdx,

ã_{i}^{j}(t, x, η, ζ) = e^{−αt}a_{i}^{j}(t, x, e^{αt}η, e^{αt}ζ),
b̃^{j}(t, x, η, ζ) = e^{−αt}b^{j}(t, x, e^{αt}η, e^{αt}ζ),
[B̃^{j}(ũ), v^{j}] = ∫_{Q_{T}^{j}} G̃^{j}(ũ)v^{j}dtdx − ∫_{S_{T}} H̃^{j}(ũ)v^{j}dtdσ_{x},

[G̃^{j}(ũ)](t, x) = e^{−αt}[G^{j}(e^{αt}ũ)](t, x), [H̃^{j}(ũ)](t, x) = e^{−αt}[H^{j}(e^{αt}ũ)](t, x). The solution of (1.31), (1.32) is unique because by (1.28) - (1.29) the operator Ã + B̃ + αI is monotone if α is sufficiently large:

[(Ã + B̃)(ũ) + αũ − (Ã + B̃)(ṽ) − αṽ, ũ − ṽ] ≥ 0.

2 Boundedness and stabilization

One can prove an existence theorem also for the interval (0, ∞). Denote by X_{∞} and X_{∞}^{⋆} the set of functions

u : (0, ∞) → V, w : (0, ∞) → V^{⋆},

respectively, such that (for their restrictions to (0, T))

u ∈ L^{p}(0, T; V), w ∈ L^{q}(0, T; V^{⋆})

for any finite T > 0. Further, let Q_{∞}^{j} = (0, ∞)×Ω_{j}, S_{∞} = [0, ∞)×S. L_{loc}^{q}(Q_{∞}^{j}) will denote the set of functions v^{j} : Q_{∞}^{j} → R such that v^{j}|_{Q_{T}^{j}} ∈ L^{q}(Q_{T}^{j}); L_{loc}^{q}(S_{∞}) will denote the set of functions v : S_{∞} → R such that v|_{S_{T}} ∈ L^{q}(S_{T}).

Theorem 2.1 Assume that we have functions a_{i}^{j}, b^{j} : Q_{∞}^{j} × R^{n+1} → R such that assumptions I - IV are satisfied for any finite T with the same constants c_{j} and functions k_{i}^{j}. Further, operators G^{j} : X_{∞} → L_{loc}^{q}(Q_{∞}^{j}) and H^{j} : X_{∞} → L_{loc}^{q}(S_{∞}) are such that their restrictions to L^{p}(0, T; V) satisfy V - VII. Assume that G^{j}, H^{j} are of Volterra type, which means that [G^{j}(u)](t, x), [H^{j}(u)](t, x) depend only on the restrictions of u^{j} to (0, t) × Ω_{j} (j = 1, 2). Then for any F ∈ X_{∞}^{⋆} there exists u ∈ X_{∞} such that the statement of Theorem 1.1 holds for any finite T.

Proof Let T_{k} be a strictly increasing sequence of positive numbers with lim(T_{k}) = +∞. For arbitrary k there exists a weak solution u_{k} ∈ L^{p}(0, T_{k}; V) of (1.9), (1.10) with T = T_{k}. Since G^{j}, H^{j} are of Volterra type, the restrictions of u_{l}^{j} to Q_{T_{k}}^{j} is a solution in Q_{T_{k}}^{j} if l > j. By using a diagonal process and arguments of the proof of Theorem 1.1 we can select a subsequence of (u_{k}) which is weakly convergent to a function u ∈ X_{∞} in L^{p}(0, T; V) for arbitrary finite T and the statement of of Theorem 1.1 holds for u with any finite T.

If some additional conditions are satisfied then one can prove that

y(t) =∥ u(t) ∥_{L^{2}(Ω_{1})×L^{2}(Ω_{2})}^{2}

is bounded in (0, ∞) for a solution u.

Theorem 2.2 Let the assumptions of Theorem 2.1 be satisfied and assume that p > 2,

∥ F (t) ∥_{V^{⋆}} is bounded , t ∈ [0, ∞),  (2.33)

for arbitrary u ∈ X_{∞}

∫_{Ω_{j}} |G^{j}(u)(t, x)|^{q}dx + ∫_{S} |H^{j}(u)(t, x)|^{q}dσ_{x} ≤  (2.34)

c_{4} sup_{[0,t]} |y| + c_{5}(t) sup_{[0,t]} |y|^{p/2} + c_{6}

where c_{4}, c_{6} are constants and c_{5} is a continuous function with lim_{∞} c_{5} = 0. Then y is bounded in [0, ∞) for a solution u. Further,

∫_{T_{1}}^{T_{2}} ∥ u(t) ∥_{V}^{p} dt ≤ c′(T_{1} − T_{2}) + c”, 0 < T_{1} < T_{2}  (2.35)

with some constants c′, c”, not depending on T_{1}, T_{2}.

The idea of the proof Applying (1.9) to u = (u^{1}, u^{2}) with arbitrary T_{1} < T_{2} we obtain

∫_{T_{1}}^{T_{2}}⟨D_{t}u^{j}(t), u^{j}(t)⟩dt + ∫_{T_{1}}^{T_{2}}⟨[A^{j}(u^{j})](t), u^{j}(t)⟩dt+  (2.36)

∫_{T_{1}}^{T_{2}}⟨[B^{j}(u^{1}, u^{2})](t), u^{j}(t)⟩dt = ∫_{T_{1}}^{T_{2}}⟨F^{j}(t), u^{j}(t)⟩dt.

Since y is absolutely continuous and

y′(t) = 2⟨D_{t}u^{1}(t), u^{1}(t)⟩ + 2⟨D_{t}u^{2}(t), u^{2}(t)⟩

(see, e.g., [8]), by using assumption IV, (2.33), (2.34), Young’s inequality and Hölder’s inequality, we obtain from (2.36) the inequality

y(T_{2}) − y(T_{1}) + c_{3}^{⋆} ∫_{T_{1}}^{T_{2}}[y(t)]^{p/2}dt ≤  (2.37)

c_{4}^{⋆} ∫_{T_{1}}^{T_{2}} [sup_{[0,t]} y + c_{5}(t) sup_{[0,t]} y^{p/2} + 1] dt

where c_{3}^{⋆}, c_{4}^{⋆} are constants. It is not difficult to show that (2.37) and p > 2 imply the boundedness of y and (2.35).

Remark 3 The estimation (2.34) is fulfilled for G^{j}, e.g. if G^{j} is given in examples a/ or b/ and the functions γ^{j} satisfy

|γ^{1}(t, x, θ_{1}, θ_{2})|^{q}, |γ^{1}(t, τ, x, θ_{1}, θ_{2})|^{q} ≤ c_{5}^{⋆}(θ_{1}^{2} + θ_{2}^{2}) + c_{6}^{⋆}(t)|θ_{2}|^{p} + c_{7}^{⋆},
|γ^{2}(t, x, θ_{1}, θ_{2})|^{q}, |γ^{2}(t, τ, x, θ_{1}, θ_{2})|^{q} ≤ c_{5}^{⋆}(θ_{1}^{2} + θ_{2}^{2}) + c_{6}^{⋆}(t)|θ_{1}|^{p} + c_{7}^{⋆},

respectively, with some constants c_{7}^{⋆}, c_{9}^{⋆}, lim_{∞} c_{6} = 0 and there is a positive number ρ such that

γ^{j}(t, τ, x, θ_{1}, θ_{2}) = 0 if τ ≤ t − ρ.

The estimation (2.34) is fulfilled for H^{j}, e.g. if H^{j} is given in examples c/, d/, e/ or f/, the functins h^{j} are bounded and

h^{j}(t, τ, x, θ_{1}, θ_{2}) = 0 if τ ≤ t − ρ.

By using monotonicity arguments, similarly to Theorem 2.2, one can prove the following stabilization result.

Theorem 2.3 Assume that the conditions of Theorem 2.2 are fulfilled and

∑_{i=1}^{n}[a_{i}^{j}(t, x, η, ζ) − a_{i}^{j}(t, x, η^{⋆}, ζ^{⋆})](ζ_{i} − ζ_{i}^{⋆})+

[b^{j}(t, x, η, ζ) − b^{j}(t, x, η^{⋆}, ζ^{⋆})](η_{i} − η_{i}^{⋆}) ≥ c[α_{j}|η − η^{⋆}|^{p} + |ζ − ζ^{⋆}|^{p}]

with some constant c > 0 and α_{1} = 1, α_{2} = 0; for a.e. x ∈ Ω_{j}, each (η, ζ) ∈ R^{n+1}

lim_{t→∞} a_{i}^{j}(t, x, η, ζ) = a_{i,∞}^{j}(x, η, ζ), lim_{t→∞} b^{j}(t, x, η, ζ) = b_{∞}^{j}(x, η, ζ),

a_{i,∞}^{j}, b_{∞}^{j} satisfy the Carathéodory condition. Further, assume that for any u ∈ X_{∞}

∫_{Ω_{j}} |G^{j}(u)(t, x)|^{q}dx + ∫_{S} |H^{j}(u)(t, x)|^{q}dσ_{x}  (2.38)

≤ c_{4}(t) sup_{[0,t]} |y| + c_{5}(t) sup_{[0,t]} |y|^{p/2} + c_{6}(t), t ∈ (0, ∞)

where

y(t) =∥ u(t) ∥_{L^{2}(Ω_{1})×L^{2}(Ω_{2})}^{2}, lim_{∞} c_{ν} = 0, ν = 4, 5, 6.

Finally, assume that there exists F_{∞} ∈ V^{⋆} such that

lim_{t→∞} ∥ F (t) − F_{∞} ∥_{V^{⋆}}= 0.

If u is a solution in (0, ∞) then there exists u_{∞} ∈ V such that

lim_{t→∞} ∥ u(t) − u_{∞} ∥_{L^{2}(Ω_{1})×L^{2}(Ω_{2})}= 0

and u_{∞} is the (unique) solution of

A_{∞}^{j}(u_{∞}^{j}) = F_{∞}^{j}

where A_{∞}^{j} is defined by

⟨A_{∞}^{j}(u_{∞}^{j}), w^{j}⟩ = ∑_{i=1}^{n} ∫_{Ω_{j}} a_{i,∞}^{j}(x, u_{∞}^{j}, Du_{∞}^{j})D_{i}w^{j}dx+

∫_{Ω_{j}} b_{∞}^{j}(x, u_{∞}^{j}, Du_{∞}^{j})w^{j}dx, w^{j} ∈ V_{j}.

Remark 4 The assumption (2.38) is satisfied for the examples a/ - f/ if

|γ^{j}(t, x, θ_{1}, θ_{2})|^{q} ≤ Φ^{⋆}(t)(θ_{1}^{2} + θ_{2}^{2}) + Φ̃(t),
|γ^{j}(t, τ, x, θ_{1}, θ_{2})|^{q} ≤ Φ^{⋆}(t)(θ_{1}^{2} + θ_{2}^{2}) + Φ̃(t),

respectively, with lim_{∞} Φ^{⋆} = lim_{∞} Φ̃ = 0 and there is a positive number ρ such that

γ^{j}(t, τ, x, θ_{1}, θ_{2}) = 0 if τ ≤ t − ρ;

further,

|h^{j}(t, x, θ_{1}, θ_{2})|^{q} ≤ Φ̃(t), |h^{j}(t, τ, x, θ_{1}, θ_{2})|^{q} ≤ Φ̃(t)

and

h^{j}(t, τ, x, θ_{1}, θ_{2}) = 0 if τ ≤ t − ρ.

References

[1] R.A Adams. Sobolev spaces. Academic Press, New York - San Francisco - London, 1975.

[2] J. Berkovits, V. Mustonen. Topological degreee for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems. Rend. Mat. Ser. VII, 12, Roma (1992), 597-621.

[3] W. Jäger, N. Kutev. Discontinuous solutions of the nonlinear transmission problem for quasilinear elliptic equations. Preprint IWR der Universität Heidelberg 98-22 (SFB 359), 1-37.

[4] J.L. Lions. Quelques métodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Villars, Paris, 1969.

[5] L. Simon. On systems of strongly nonlinear parabolic functional differential equations. Periodica Math. Hung. 33 (1996), 135-151.

[6] L. Simon. On different types of nonlinear parabolic functional differential equations. Pure Math. Appl. 9 (1998), 181-192.

[7] W. Jäger, L. Simon. On transmission problems for nonlinear parabolic differential equations. Annales Univ. Sci. Budapest 45 (2002), 143-158.

[8] E. Zeidler. Nonlinear functional analysis and its applications II A and II B. Springer, 1990.

(Received September 23, 2003)

László Simon Department of Applied Analysis L. Eötvös University of Budapest Pázmány Péter sétány 1/C H-1117 Budapest Hungary

simonl@ludens.elte.hu