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.
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:
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 utt. 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
with initial condition
and boundary condition of the Dirichlet type
We will assume that u0 ∈ H2(Ω) and Ω ⊆ ℝN 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
(A2) There exists a constant c̄ > 0, such that
with certain exponent
(A3) There exist constants c1, c2 >0 and exponents
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,
(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
There exist constant L1 >0 and exponents
Putting u2 = 0 to (4) and using (A5) we obtain:
When we note that index r1 + 1 is no greater than q we observe that assumption (A3) holds. Similarly, as a consequence of (4)
Taking the limit with u2 → u1 we obtain that
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
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
with initial – boundary conditions:
We will search a fixed point of the operator F(·, 1) in
The existence of the solution of the problem (1) is equivalent to the existence of the fixed point of operator F(·, 1) in
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
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
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:
Proof
Multiplying the equation (5) by u and integrating over Ω we obtain:
Then from (A1):
Integrating first right hand side component by parts we obtain:
Using the Poincaré inequality ∫Ω |u|2 dx ≤ p ∫Ω |∇u|2 dx, we have next:
so that
Choosing
Finally using Gronwall inequality (see [10], p. 35):
for t ∈ (0, T). □
Lemma 3.2
There exist constants B, D ∈ ℝ such that:
Proof
Multiplying equation (5) by Δu and integrating over Ω:
Integrating by parts:
Then using Cauchy inequality with
and
Since
Consequently, our resulting estimate has the form:
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]):
for some constants h1, h2 > 0. Thus
Due to lemma 3.1,
Using the Young inequality with ϵ1
with a positive constant a = a(c̄, c2, ϵ1), and the exponent
Since there exists a constant č >0 such that
Therefore
and the right side is a constant. Using Gronwall inequality (see [10], p. 35), denoting B = 2(ϵ1 + 1)h1č and D = 2abP + 2c̄2|Ω| + 2h2e, we obtain:
□
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:
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
Proof
By applying the Laplace operator Δ to (5), multiplying the result by Δu and integrating over Ω we obtain:
Integrating by parts and using (A5):
Then thanks to the Cauchy inequality
Now, using assumption (A3), we will estimate last integral:
Next the norms ∥ · ∥L4s1(Ω), ∥ · ∥w1,4(Ω) and ∥ · ∥L2s2(Ω) are estimated using Gagliardo-Nierenberg inequality. Since
Due to Remark 3.3, ∥u∥H1(Ω) < M, inequality (7) will take the form:
Choosing θ = max(2s1θ1 + 2θ2, 2s2θ3) we find that θ < 2. Then, defining
and
we obtain
Estimate (6) will be extended to:
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:
Since θ < 2, we can find a constant
Then we have:
Now, using the Calderon-Zygmund type inequality (see [12], pp. 186-187):
we can find positive constants h1, h0 such that:
Using the Gronwall inequality (see [10], p. 35) we finally obtain:
for t ∈ (0, T). Then defining
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
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
and because an element υ ∈ A, Δu ∈ L2(Ω) and thanks to (A2) the function g ∈ L2(Ω), we can deduce that ut ∈ 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
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
Let us take α ∈ [0,1] and
Subtracting the above equations, defining u: = u1 − u2, using (A4), it can be seen that:
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
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
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:
Then using Cauchy inequality and statement of Lemma 3.4 we obtain:
Analogously, using (A4) and the fact that H2(Ω) ⊆ L2q1(Ω) ∩ L2q2(Ω):
for some ϵ > 0. Integrating by parts component ∫Ω · |u| dx we obtain:
Using the Poincaré inequality ∫Ω|u|2 dx ≤ p ∫Ω |∇u|2 dx, we have next:
so that
where
and
Finally, using the Gronwall inequality (see [10], p. 35):
for t ∈ (0, T). Noting that û0 = 0, we obtain:
Let us fix ε > 0 and take δ > 0 and ϵ > 0 such that
Then:
□
Continuity of the operator F(υ, α) with respect to the parameter α will be verified in the similar way. Let
Let us take α1, α2 ∈ [0,1] and assume that u1, u2 will be solutions of the problem (5), i.e.
Then, after subtracting equations and defining u = u1 − u2, we receive
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
Acknowledgement
The author is grateful to the referee for valuable remarks improving the original version of the paper.
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