The quasilinear parabolic kirchhoff equation

Łukasz Dawidowski

doi:10.1515/math-2017-0036

Artikel Open Access

The quasilinear parabolic kirchhoff equation

Łukasz Dawidowski

Veröffentlicht/Copyright: 13. April 2017

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 15 Heft 1

Abstract

In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.

Keywords: Kirchhoff equation; Quasi-linear parabolic equation

MSC 2010: 35K55; 35K15

1 Introduction

G. Kirchhoff in [1] proposed the hyperbolic integro-differential equation in order to describe small, transversal vibrations of an elastic string of length l (at rest) when the longitudinal motion can be considered negligible with respect to the transversal one.

In their papers M. Gobbino [2] and M. Nakao [3] considered some generalized degenerate Kirchhoff equations. M. Gobbino studied the equation:

ut−(1+||∇u||L2(Ω)2)Δu=0,

but his method does not use fixed point theorems and can not be applied to the problem considered in this article. In another papers M. Ghisi and M. Gobbino [4, 5] showed certain connections between the above equation and equation of hyperbolic type containing term u_tt. However M. Nakao proved the existence of solutions of the equation of hyperbolic type. We will investigate a quasilinear parabolic generalization of the Kirchhoff equation.

The proof of the existence of solution of problem considered in this paper, which is indeed quasilinear (i.e. the derivative of solution is a part of coefficient of the main part), can not be carried out using most classical methods. This paper is devoted to this proof.

Consider the Dirichlet problem for quasilinear generalized degenerate Kirchhoff equation

(1) ut−(1+||∇u||L2(Ω)2)Δu+g(u,x)=0

with initial condition

(2) u(0,x)=u0(x),x∈Ω,

and boundary condition of the Dirichlet type

(3) u|∂Ω=0.

We will assume that u₀ ∈ H²(Ω) and Ω ⊆ ℝ^N is a domain of the class C².

The following conditions will be imposed on the nonlinear function g: ℝ × Ω → ℝ throughout the paper:

(A1) There exists a function d:Ω → ℝ, such that ∫_Ω d(x) dx = d < ∞ and a constant c > 0, that

−g(u,x)u≤cu2+d(x).

(A2) There exists a constant c̄ > 0, such that

|g(u,x)|≤c¯(1+|u|q),

with certain exponent q≤N+2N .

(A3) There exist constants c₁, c₂ >0 and exponents s1∈(0,4N−2),s2∈(0,N+4N−2) such that:

| ∂g∂u |≤c1(1+|u|s1) and | ∂g∂xi |≤c2(1+|u|s2).

In dimension N = 2 we assume only that s₁, s₂ > 0.

(A4) The function g is locally Lipschitz continuous with respect to the first variable, i.e. there exist constants L > 0, q1,q2∈(0,NN−4) (or if N ≤ 4, then q₁, q₂ ∈ ℝ), such that

|g(u1,x)−g(u2,x)|≤L|u1−u2|(1+|u1|q1+|u2|q2).

(A5) g(0, x) = 0 for all x ∈ Ω.

Remark 1.1

Instead of assuming (A2), (A4) and the first part of (A3) (i.e. there exist constant c₁ >0 and s1∈(0,4N−2) such that ∂g∂u≤c1(1+|u|s1) ) we can assume that:

There exist constant L₁ >0 and exponents r1,r2∈(0,2N) such that

(4) |g(u1,x)−g(u2,x)|≤L1|u1−u2|(1+|u1|r1+|u2|r2).

Putting u₂ = 0 to (4) and using (A5) we obtain:

|g(u1,x|=|g(u1,x)−g(0,x)|≤L1|u1|(1+|u1|r1).

When we note that index r₁ + 1 is no greater than q we observe that assumption (A3) holds. Similarly, as a consequence of (4)

|g(u1,x)−g(u2,x)||u1−u2|≤L2(1+|u1|r1+|u2|r2).

Taking the limit with u₂ → u₁ we obtain that |∂g∂u|≤L2(1+|u1|r1+|u1|r2) and when we put s₁ := max(r₁, r₂), then (A4) holds.

Constructing a solution of (1) the Leray – Schauder Principle will be used (see e.g. [6], p. 189). We recall it here for completeness of the presentation:

Proposition 1.2

(Leray – Schauder Principle). Consider a transformation y = T(x, k) where x, y belong to a Banach space X and k is a real parameter which varies in a bounded interval, say a ≤ k ≤ b. Assume that

T(x, k) is defined for all x ∈ X and a ≤ k ≤ b,
for any fixed k, T(x, k) is continuous as a function of x, i.e. for any x₀ ∈ X and for any ε > 0 there exists a δ > 0 such that ||T(x, k) − T(x₀, k)|| < ε if ||x − x₀|| < δ,
for x varying in bounded set in X, T(x, k) is uniformly continuous in k, i.e. for any bounded set X₀ ⊆ X and for any ε > 0 there exists a δ > 0 such that if x ∈ X₀, |k₁ − k₂| < δ, a ≤ k₁, k₂ ≤ b, then ∥T(x, k₁) − T(x, k₂)∥ < ε
for any fixed k, T(x, k) is a compact transformation, i.e. it maps bounded subsets of X into compact subsets of X,
there exists a (finite) constant M such that every possible solution X of x − T(x, k) = 0 (x ∈ X, a ≤ k ≤ b) satisfies: ∥x∥ < M,
the equation x − T(x, a) = y has the unique solution for any y ∈ X.

Then there exists a solution of the equation x − T(x, b) = 0.

Assumption (f) means that Leray-Schauder degree

degLS(I−T(⋅,a);B(0,M);0)≠0

with the constant M which comes from assumption (e). The more standard version of Leray-Schauder Principle, called also Leray-Schauder continuation theorem, can be found e.g. in [7], p. 351 (Theorem 13.3.7).

2 Main theorem

Let us fix arbitrary T > 0.

We introduce an operator F : L ∞ ( [ 0 , T ] , H 0 1 ( Ω ) ) × [ 0 , 1 ] → L ∞ ( [ 0 , T ] , H 0 1 ( Ω ) ) in such a way, that for every function υ ∈ L ∞ ( [ 0 , T ] ) , H 0 1 ( Ω ) ) and α ∈ [0, 1], u = F(υ, α) is a solution of the equation

(5) ut−(1+α‖∇υ‖L2(Ω)2)Δu+g(u,x)=0,

with initial – boundary conditions:

u(0,x)=u0(x),x∈Ω,u|∂Ω=0.

We will search a fixed point of the operator F(·, 1) in L∞([0,T],H01(Ω)) .

The existence of the solution of the problem (1) is equivalent to the existence of the fixed point of operator F(·, 1) in L∞([0,T],H01(Ω)) .

We have:

Theorem 2.1

Under the assumptions (A1), (A2), (A3), (A4) and (A5) there exists a solution of the problem (1) with initial-boundary conditions (2), (3) in the space L∞([0,T],H01(Ω)) .

It can be seen that, using standard theory (see e.g. [8], chapter 3 for details), for α = 0 the equation u − F(u, 0) = y has a unique solution for any y∈L∞([0,T]),H01(Ω)) ; equivalently, the semilinear heat equation

ut−Δu+g(u,x)=0

with Dirichlet boundary condition has a unique solution.

The proof of the theorem will be given in a few steps. We start with obtaining certain a priori estimates.

3 Some lemmas

First it can be mentioned that when u₀ ∈ H²(Ω) then, using the method of Tanabe and Sobolevski (see [9], page 438), the solution of the problem (5) varies in the space H²(Ω).

Lemma 3.1

There exists A ∈ ℝ that for all t ∈ (0, T) this estimate holds:

∫ Ω u 2 d x ≤ e 2 A t ∫ Ω u 0 2 d x + d A − d A .

Proof

Multiplying the equation (5) by u and integrating over Ω we obtain:

∫Ωu⋅ut dx−(1+α‖∇υ‖L2(Ω)2∫ΩΔu⋅u dx+∫Ωg(u,x)u dx=0.

Then from (A1):

12ddt∫Ω|u|2dx≤(1+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅u+∫Ω(cu2+d(x)) dx.

Integrating first right hand side component by parts we obtain:

12ddt∫Ω|u|2dx≤−(1+α‖∇υ‖L2(Ω)2)∫Ω|∇u|2dx+c∫Ωu2dx+d.

Using the Poincaré inequality ∫_Ω |u|² dx ≤ p ∫_Ω |∇u|² dx, we have next:

12ddt∫Ω|u|2dx≤−1p(1+α‖∇υ‖L2(Ω)2)∫Ω|u|2dx+x∫Ωu2dx+d,

so that

12ddt∫Ω|u|2dx≤[c−1p(1+α‖∇υ‖L2(Ω)2)]∫Ω|u|2dx+d.

Choosing A=c−1p the estimates holds

12ddt∫Ω|u|2dx≤A∫Ω|u|2dx+d.

Finally using Gronwall inequality (see [10], p. 35):

∫Ωu2dx≤e2At(∫Ωu02 dx+dA)−dA,

for t ∈ (0, T). □

Lemma 3.2

There exist constants B, D ∈ ℝ such that:

∫Ω|∇u|2dx≤eBt(∫Ω|∇u0|2dx−DB)+DB.

Proof

Multiplying equation (5) by Δu and integrating over Ω:

∫ΩΔu⋅ut dx−(1+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅Δu dx+∫Ωg(u,x)⋅Δu dx.

Integrating by parts:

∫ΩΔu⋅ut dx=12ddt∫Ω|∇u|2dx.

Then using Cauchy inequality with ε=12 and (A2):

12ddt∫Ω|∇u|2dx=−(1+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅Δu dx+∫Ωg(u,x)⋅Δu dx ≤ ≤−(1+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅Δu dx+12∫Ω(g(u,x))2dx+12∫Ω(Δu)2dx≤ ≤−(12+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅Δu dx+12c¯2∫Ω(1+|u|q)2dx.

and

12ddt∫Ω|∇u|2dx≤−(12+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅Δu dx+12c¯2∫Ω(1+|u|q)2dx≤ ≤−(12+α‖∇υ‖L2(Ω)2)∫Ω|Δu|2dx+c¯2∫Ω(1+|u|2q) dx= =−(12+α‖∇υ‖L2(Ω)2)∫Ω|Δu|2dx+c¯2|Ω|+c¯2∫Ω|u|2qdx.

Since q<N+2N when θ=N4(1−1q) , the following estimate holds:

‖ϕ‖L2q(Ω)2q≤c2‖ϕ‖L2(Ω)2q(1−θ)⋅‖ϕ‖H2(Ω)2qθ and 2qθ<1.

Consequently, our resulting estimate has the form:

12ddt∫Ω|∇u|2dx≤−(12+α‖∇υ‖L2(Ω)2)∫Ω(Δu)2dx+c¯2|Ω|+c¯2c2‖u‖L2(Ω)2q(1−θ)⋅‖u‖H2(Ω)2qθ

Because the norms ∥ · ∥_H²(Ω) and ∫_Ω ·² dx + ∫_Ω(Δ·)² dx are equivalent on the domain of (−Δ) (for more details see e.g. [11]):

(12+α‖∇υ‖L2(Ω)2)∫Ω(Δϕ)2dx≥h1‖ϕ‖H2(Ω)2−h2∫Ωϕ2dx,

for some constants h₁, h₂ > 0. Thus

12ddt∫Ω|∇u|2dx≤−[h1‖u‖H2(Ω)2−h2∫Ωu2dx] c¯2|Ω|+c¯2c2‖u‖L2(Ω)2q(1−θ)⋅‖u‖H2(Ω)2qθ= =−h1‖u‖H2(Ω)2+c¯2c2‖u‖L2(Ω)2q(1−θ)⋅‖u‖H2(Ω)2qθ+c¯2|Ω|+h2∫Ωu2dx.

Due to lemma 3.1, supt∈[0,T]∫Ωu2dx≤e=e(c,d,T) , so that the estimate holds

12ddt∫Ω|∇u|2dx≤−h1‖u‖H2(Ω)2+c¯2c2‖u‖L2(Ω)2q(1−θ)+c¯2|Ω|+h2e.

Using the Young inequality with ϵ₁

c¯2c2‖u‖L2(Ω)2q(1−θ)⋅‖u‖H2(Ω)2qθ≤ϵ1h1‖u‖H2(Ω)2+a‖u‖L2(Ω)P,

with a positive constant a = a(c̄, c₂, ϵ₁), and the exponent P=11−qθ is chosen in such a way that Young inequality holds. Then:

12ddt∫Ω|∇u|2dx≤−(ϵ1+1) h1‖u‖H2(Ω)2+a‖u‖L2(Ω)P+c¯2|Ω|+h2e.

Since there exists a constant č >0 such that u H 2 ( Ω ) 2 ≥ C ˇ ∫ Ω ∇ u 2 d x and ‖u‖L2(Ω)2≤b<∞ then:

1 2 d d t ∫ Ω ∇ u 2 d x ≤ − ( ϵ 1 + 1 ) h 1 c ˇ ∫ Ω ∇ u 2 d x + a b P + c − 2 Ω + h 2 e .

Therefore

d d t ∫ Ω ∇ u 2 d x + 2 ( ϵ 1 + 1 ) h 1 c ˇ ∫ Ω ∇ u 2 d x ≤ 2 a b P + 2 c ¯ 2 Ω + 2 h 2 e

and the right side is a constant. Using Gronwall inequality (see [10], p. 35), denoting B = 2(ϵ₁ + 1)h₁č and D = 2ab^P + 2c̄²|Ω| + 2h₂e, we obtain:

∫Ω|∇u|2dx≤eBt(∫Ω|∇u0|2dx−DB)+DB.

□

Remark 3.3

The two previous lemmas provide us an a priori estimate of the solution u of (5) in the space ℒ^∞([0, T],H¹(Ω)).

Let us take a constant M > 0 such that:

‖u‖L∞([0,T],H1(Ω))=supt∈[0,T]‖u(t)‖H1(Ω)<M

The lemmas show also that, if u is a fixed point of the operator F, its norm ∥u∥_{ℒ^∞([0,T],H¹(Ω))} will be bounded by M, since the constants in both lemmas are independent of u and α.

Finally a third a priori estimation in H²(Ω) will be shown:

Lemma 3.4

There exists a constant M₁ > 0 such that

‖Δu‖L2(Ω)≤M1<∞,

Proof

By applying the Laplace operator Δ to (5), multiplying the result by Δu and integrating over Ω we obtain:

∫ΩΔu⋅Δut dx−(1+α‖∇υ‖L2(Ω)2∫ΩΔu⋅Δ2u dx+∫Ωg(u,x)Δ2u dx=0.

Integrating by parts and using (A5):

12ddt∫Ω|Δu|2dx+(1+α‖∇υ‖L2(Ω)2∫Ω|∇Δu|2dx=∫Ω∇g(u,x)(∇Δu) dx.

Then thanks to the Cauchy inequality

(6) 12ddt∫Ω|Δu|2dx+(12+α‖∇υ‖L2(Ω)2)∫Ω|∇Δu|2dx≤12∫Ω|∇g(u,x)|2dx.

Now, using assumption (A3), we will estimate last integral:

(7) 12∫Ω|∇g(u,x)|2dx≤∫Ω∑i=1N(∂g∂u)2(∂u∂xi)2dx+∫Ω∑i=1N(∂g∂xi)2dx≤ ≤∑i=1N∫Ω(∂g∂u)4dx⋅∫Ω(∂u∂xi)4dx+∑i=1N∫Ω(∂g∂xi)2dx≤ ≤N⋅c1⋅∫Ω(1+|u|4s1)dx⋅‖u‖W1,4(Ω)2+N⋅c2⋅∫Ω(1+|u|2s2) dx ≤ ≤(c1N|Ω|+c1N‖u‖L4s1(Ω)2s1)⋅‖u‖W1,4(Ω)2+c2N|Ω|+c2N|u|L2s2(Ω)2s2

Next the norms ∥ · ∥_L^4s₁(Ω), ∥ · ∥_w^1,4(Ω) and ∥ · ∥_L^2s₂(Ω) are estimated using Gagliardo-Nierenberg inequality. Since s1∈(0,4N−2),s2∈(0,N+4N−2) it is possible to find constants c₃, c₄, c₅ > 0, and powers θ₁, θ₂, θ₃ ∈ (0,1), such that:

‖u‖L4s1(Ω)≤c3‖u‖H3(Ω)θ1‖u‖H1(Ω)1−θ1,‖u‖W1,4(Ω)≤c4‖u‖H3(Ω)θ2‖u‖H1(Ω)1−θ2, 2s1θ1+2θ2<2,‖u‖L2s2(Ω)≤c5‖u‖H3(Ω)θ3‖u‖H1(Ω)1−θ3, 2s2θ3<2.

Due to Remark 3.3, ∥u∥_H¹(Ω) < M, inequality (7) will take the form:

(8) 1 2 ∫ Ω ∇ g ( u , x ) 2 d x ≤ c 1 N Ω + c 1 N u L 4 s 1 ( Ω ) 2 s 1 ⋅ u W 1 , 4 ( Ω ) 2 + + c 2 N Ω + c 2 N u L 2 s 2 ( Ω ) 2 s 2 ≤ ≤ c 1 N Ω + c 1 N c 3 M 2 s 1 ( 1 − θ 1 ) u H 3 ( Ω ) 2 s 1 θ 1 ⋅ ⋅ c 4 M 2 ( 1 − θ 2 ) u H 3 ( Ω ) 2 θ 2 + + c 2 N Ω + c 2 N c 5 M 2 s 2 ( 1 − θ 3 ) u H 3 ( Ω ) 2 s 2 θ 3 .

Choosing θ = max(2s₁θ₁ + 2θ₂, 2s₂θ₃) we find that θ < 2. Then, defining

const=c2N|Ω|

and

k=max(c1N|Ω|c4M2(1−θ2),c1Nc3M2s1(1−θ1)c4M2(1−θ2),c2Nc5M2s2(1−θ3),

we obtain

12∫Ω|∇g(u,x)|2dx≤const+k‖u‖H3(Ω)θ.

Estimate (6) will be extended to:

12ddt∫Ω|Δu|2dx+(12+α‖∇υ‖L2(Ω)2)∫Ω|∇Δu|2dx≤const+k‖u‖H3(Ω)θ.

Now, since the norms ∥ · ∥_H³(Ω) and ∥ · ∥_L²(Ω) + ∥∇ · ∥_L²(Ω) + ∥ Δ · ∥_L²(Ω) + ∥∇Δ · ∥_L²(Ω) are equivalent, and due to remark 3.3, we have:

∃c¯>0‖u‖H3(Ω)≤c¯(M+‖∇Δu‖L2(Ω)).

Since θ < 2, we can find a constant c^<12 such that:

k‖u‖H3(Ω)θ≤c^‖∇Δu‖L2(Ω)2+const=c^∫Ω|∇Δu|2dx+const

Then we have:

ddt∫Ω|Δu|2dx+α‖∇υ‖L2(Ω)2∫Ω∇Δu dx≤const,

Now, using the Calderon-Zygmund type inequality (see [12], pp. 186-187):

∃c¯>0∀ϕ∈D(−Δ3/2)∫Ω|Δϕ|2dx≤c˜∫Ω|∇Δϕ|2dx.

we can find positive constants h₁, h₀ such that:

ddt∫Ω|Δu|2dx+h1∫Ω|Δu|2dx≤h2,

Using the Gronwall inequality (see [10], p. 35) we finally obtain:

∫Ω|Δu|2dx≤eh1t(∫Ω(Δu0)2dx−h2h1)+h2h1

for t ∈ (0, T). Then defining M1=supt∈(0,T)[eh1t(∫Ω(Δu0)2dx−h2h1)+h2h1] the proof will be completed. □

Remark 3.5

Above lemmas show that every eventual solution in the sense of Theorem 2.1 has to be an element of the space L∞([0,T],H2(Ω)∩H01(Ω)) .

4 Proof of the main theorem

This section is devoted to the proof of the Theorem 2.1. Three conditions from the Leray – Schauder Principle: continuities (b), (c) and compactness (d) will be verified.

It has to be proved that the operator F(·, α) is compact, i.e. it maps bounded subsets of L∞([0,T],H01(Ω)) into compact subsets of L∞([0,T],H01(Ω)) . Let us fix α ∈ [0,1]. If the bounded subset A of L∞([0,T],H01(Ω)) is taken, then F(A, α) is bounded in the space L∞([0,T],H2(Ω)∩H01(Ω)) . Using the equation (5), we can write:

ut=(1+α‖∇υ‖L2(Ω)2)Δu−g(u,x)

and because an element υ ∈ A, Δu ∈ L²(Ω) and thanks to (A2) the function g ∈ L²(Ω), we can deduce that u_t ∈ L²(Ω). Additionally, the embedding H²(Ω) ⊆ H¹(Ω) is compact and H¹(Ω) ⊆ L²(Ω) is continuous. Using Aubin lemma (see e.g. [13] and [14]) the set of values of operator F(A, α) is compact in L∞([0,T],H01(Ω)) .

Now we prove that for any fixed α ∈ [0, 1], the operator F(υ, α) is continuous as a function of υ, i.e. for any υ1∈L∞([0,T],H01(Ω)) and for any ε > 0 there exists a δ > 0 such that ∥F(υ₁, α) − F(υ₂, α)∥ < ε if ∥υ₁ − v₂∥ < δ.

Let us take α ∈ [0,1] and υ1, υ2∈L∞([0,T],H01(Ω)) . Let u1,u2∈L∞([0,T],H01(Ω)) be the values of the operator F corresponding to υ₁ and υ₂, i.e.

( u 1 ) t − ( 1 + α ∇ υ 1 L 2 ( Ω ) 2 ) Δ u 1 + g ( u 1 , x ) = 0 , ( u 2 ) t − ( 1 + α ∇ υ 2 L 2 ( Ω ) 2 ) Δ u 2 + g ( u 2 , x ) = 0.

Subtracting the above equations, defining u: = u₁ − u₂, using (A4), it can be seen that:

(9) ut−(1+α‖∇υ1‖L2(Ω)2)Δu≤α⋅(Δu2)⋅‖υ1−υ2‖L∞([0,T],H1(Ω))+L⋅u⋅(1+|u1|q1+|u2|q2).

Then we have that u(0, x) = û₀ = 0 for all x ∈ Ω. Analogously as in the lemmas above, we see that for all ε > 0 there exists some δ > 0 such that

‖υ1−υ2‖L∞([0,T],H01(Ω))<δ ⇒ ‖u1−u2‖L∞([0,T],H1(Ω))<ε.

As an example we prove the estimation for ∥u∥L²(Ω). Similarly, we can prove the estimations for ∥∇u∥_L²(Ω), ∥Δu∥_L²(Ω).

Lemma 4.1

If function u∈L∞([0,T],H01(Ω)) is a solution of inequality (9) with initial condition u(0, x) = 0 for x ∈ Ω, then for all ε > 0 there exists some δ > 0 such that if ‖υ1−υ2‖L∞([0,T],H01(Ω))<δ then

‖u‖L∞([0,T],L2(Ω))=‖u1−u2‖L∞([0,T],L2(Ω))<ε.

Proof

The proof is similar to the proof of Lemma 3.1. By multiplying the inequality (9) by |u| and integrating over Ω we obtain:

∫Ω|u|⋅ut dx−(1+α‖∇υ‖L2(Ω)2)∫ΩΔu⋅|u|dx≤ ≤α⋅‖υ1−υ2‖L∞([0,T],H1(Ω))∫Ω|u|⋅(Δu2)dx +L⋅∫Ωu2⋅(1+|u1|q1+|u2|q2) dx.

Then using Cauchy inequality and statement of Lemma 3.4 we obtain:

∫Ω|u|⋅(Δu2) dx≤12∫Ωu2dx+12∫Ω(Δu2)2dx≤12∫Ωu2dx+12M2.

Analogously, using (A4) and the fact that H²(Ω) ⊆ L^2q₁(Ω) ∩ L^2q₂(Ω):

∫ Ω u 2 ⋅ ( 1 + u 1 q 1 + u 2 q 2 ) d x ≤ 1 2 ϵ ∫ Ω u 2 d x + 2 ϵ ∫ Ω ( 1 + u 1 2 q 1 + u 2 2 q 2 ) d x ≤ ≤ 1 2 ϵ ∫ Ω u 2 d x + 2 ϵ ( Ω + 2 M 2 ) ≤ 1 2 ∫ Ω u 2 d x + ϵ C 1 .

for some ϵ > 0. Integrating by parts component ∫_Ω · |u| dx we obtain:

1 2 d d t ∫ Ω u 2 d x + ( 1 + α ∇ υ L 2 ( Ω ) 2 ) ∫ Ω ∇ u 2 d x ≤ ≤ α ⋅ υ 1 − υ 2 L ∞ ( [ 0 , T ] , H 1 ( Ω ) ) 1 2 ∫ Ω u 2 d x + 1 2 M 2 + 1 2 ϵ ⋅ L ∫ Ω u 2 d x + L ⋅ ϵ ⋅ C 1 .

Using the Poincaré inequality ∫_Ω|u|² dx ≤ p ∫_Ω |∇u|² dx, we have next:

1 2 d d t ∫ Ω u 2 d x + 1 p ( 1 + α ∇ υ L 2 ( Ω ) 2 ) ∫ Ω u 2 d x ≤ ≤ α ⋅ υ 1 − υ 2 L ∞ ( [ 0 , T ] , H 1 ( Ω ) ) 1 2 ∫ Ω u 2 d x + 1 2 M 2 + 1 2 ϵ ⋅ L ∫ Ω u 2 d x + L ⋅ ϵ ⋅ C 1 .

so that

d d t ∫ Ω u 2 d x ≤ C ^ ∫ Ω u 2 d x + C ^ 1 .

where

C^=[α⋅‖υ1−υ2‖L∞([0,T],H1(Ω))+Lϵ−2p(1+α‖∇υ‖L2(Ω)2)]

and

C ^ 1 = α ⋅ υ 1 − υ 2 L ∞ ( [ 0 , T ] , H 1 ( Ω ) ) ⋅ M 2 + L ⋅ ϵ ⋅ C 1 .

Finally, using the Gronwall inequality (see [10], p. 35):

∫Ωu2dx≤eC^t(∫Ωu^02 dx+C^1C^)−C^1C^,

for t ∈ (0, T). Noting that û₀ = 0, we obtain:

∫Ωu2dx≤C^1C^eC^t−C^1C^,

Let us fix ε > 0 and take δ > 0 and ϵ > 0 such that

‖υ1−υ2‖L∞([0,T],H1(Ω))<δC^1=α⋅‖υ1−υ2‖L∞([0,T],H1(Ω))⋅M2+L⋅ϵ⋅C1<δδC^ supt∈(0,T)eC^t−δC^<ε.

Then:

∫Ωu2dx≤δC^⋅ supt∈(0, T)eC^t−δC^<ε.

□

Continuity of the operator F(υ, α) with respect to the parameter α will be verified in the similar way. Let X⊆L∞([0,T],H01(Ω)) be a bounded subset and υ ∈ X. Then there exists a constant N > 0 such that

‖υ‖L∞([0,T],H01(Ω))≤N, for υ∈X

Let us take α₁, α₂ ∈ [0,1] and assume that u₁, u₂ will be solutions of the problem (5), i.e.

( u 1 ) t − ( 1 + α 1 ∇ υ L 2 ( Ω ) 2 ) Δ u 1 + g ( u 1 , x ) = 0 , ( u 2 ) t − ( 1 + α 2 ∇ υ L 2 ( Ω ) 2 ) Δ u 2 + g ( u 2 , x ) = 0.

Then, after subtracting equations and defining u = u₁ − u₂, we receive

ut−(1+α2‖∇υ‖L2(Ω)2)Δu≤N⋅(Δu2)⋅|α1−α2|+L⋅u⋅(1+|u1|q1+|u2|q2).

Because |α₁ − α₂| < δ, the continuity of the operator F with respect to a will be obtained in a similar way as above.

Remark 4.2

As a result of Theorem 2.1 and Remark 3.5 there exists a solution u ∈ ℒ^∞([0, T], H²(Ω)) of the problem (1) with initial-boundary conditions (2), (3).

In this paper the existence of a solution of some quasilinear parabolic generalization of the Kirchhoff equation of the form (1) with initial – boundary condition was proved. Using the Leray-Schauder Principle we obtain the solution in the space L∞([0,T],H2(Ω)∩H01(Ω)) for each arbitrary T > 0. Its higher regularity will be studied next.

Acknowledgement

The author is grateful to the referee for valuable remarks improving the original version of the paper.

References

[1] Kirchhoff G., Vorlesungen über Mechanik, Teubner, Stuttgart, 1883.Suche in Google Scholar

[2] Gobbino M., Quasilinear Degenerate Parabolic Equations of Kirchhoff Type, Math. Meth. Appl. Sci., 1999, 22, 375-388.10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7Suche in Google Scholar

[3] Nakao M., An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 2009, 353, 652-659.10.1016/j.jmaa.2008.09.010Suche in Google Scholar

[4] Ghisi M., Gobbino M., Hyperbolic-parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation, Math. Ann., 2012, 354, 1079-1102.10.1007/s00208-011-0765-xSuche in Google Scholar

[5] Ghisi M., Gobbino M., Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates, J. Differential Equations, 2008, 245, 2979-3007.10.1016/j.jde.2008.04.017Suche in Google Scholar

[6] Friedman A., Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.Suche in Google Scholar

[7] Granas A., Dugundji J., Fixed Point Theory, Springer Monographs in Mathematics, Springer, 2003.10.1007/978-0-387-21593-8Suche in Google Scholar

[8] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer, 1981.10.1007/BFb0089647Suche in Google Scholar

[9] Yosida K., Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1974.10.1007/978-3-642-96208-0Suche in Google Scholar

[10] Yagi A., Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, Springer Verlag, Heidelberg-Dordrecht-London-New York, 2009.10.1007/978-3-642-04631-5Suche in Google Scholar

[11] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.Suche in Google Scholar

[12] Gilbarg D., Trudinger N., Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Springer-Verlag, Berlin-Heidelberg, 2001.10.1007/978-3-642-61798-0Suche in Google Scholar

[13] Aubin J.P, Un theoreme de compacite, C. R. Acad. Sci. Paris, 1963, 256, 5042-5044.Suche in Google Scholar

[14] Amann H., Compact embeddings of vector – valued Sobolev and Besov spaces, 2000, 35 (55), 161-177.Suche in Google Scholar

Received: 2015-5-11

Accepted: 2015-11-3

Published Online: 2017-4-13

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.