Abstract
We consider the semilinear Petrovsky equation
in a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given.
Mathematics Subject Classification (2000): 35L35; 35L75; 37B25.
Keywords:
viscoelasticity; existence; blowup; lifespan; negative initial energy; positive initial energy1 Introduction
In this article, we concerned with the problem
where Ω ⊂ R^{n }is a bounded domain with smooth boundary ∂Ω in order that the divergence theorem can be applied. ν is the unit normal vector pointing toward the exterior of Ω and p > 0. Here, g represents the kernel of the memory term satisfying some conditions to be specified later.
In the absence of the viscoelastic term, i.e., (g = 0), we motivate our article by presenting some results related to initialboundary value Petrovsky problem
Research of global existence, blowup and energy decay of solutions for the initial boundary value problem (1.2) has attracted a lot of articles (see [14] and references there in).
Amroun and Benaissa [1] investigated (1.2) with f(u, u_{t}) = bu^{p2}uh(u_{t}) and proved the global existence of solutions by means of the stable set method in combined with the FaedoGalerkin procedure. In [3], Messaoudi studied problem (1.2) with f(u, u_{t}) = bu^{p2}uau_{t}^{m2}u_{t}. He proved the existence of a local weak solution and showed that this solution blows up in finite time with negative initial energy if p > m.
In the presence of the viscoelastic terms, Rivera et al. [5] considered the plate model:
in a bounded domain Ω ⊂ R^{N }and showed that the energy of solution decay exponentially provided the kernel function also decay exponentially. For more related results about the existence, finite time blowup and asymptotic properties, we refer the reader to [516].
In the present article, we devote our study to problem (1.1). We will prove the existence of weak solutions under some appropriate assumptions on the function g and blowup behavior of solutions. In order to obtain the existence of solutions, we use the FaedoGalerkin method and to get the blowup properties of solutions with nonpositive and positive initial energy, we modify the method in [17]. Estimates for the blowup time T* are also given.
2 Preliminaries
We define the energy function related with problem (1.1) is given by
where
We denote by ∥.∥_{k}, the L^{k}norm over Ω. In particular, the L^{2}norm is denoted ∥.∥_{2}. We use the familiar function spaces and throughout this article we assume and .
In the sequel, we state some hypotheses and three wellknown lemmas that will be needed later.
(A1) p satisfies
(A2) g is a positive bounded C^{1 }function satisfying g(0) > 0, and for all t > 0
also there exists positive constants L_{0}, L_{1 }such that
(A3)
Lemma 1 (SobolevPoincare's inequality). Let p be a number that satisfies (A1), then there is a constant C_{* }= C(Ω, p) such that
Lemma 2 [4]. Let δ > 0 and B(t) ∈ C^{2}(0, ∞) be a nonnegative function satisfying
If
with , then B'(t) > K_{0 }for t > 0, where K_{0 }is a constant.
Lemma 3 [4]. If Y(t) is a nonincreasing function on [t_{0}, ∞) and satisfies the differential inequality
where a > 0, δ > 0 and b ∈ R, then there exists a finite time T* such that
Upper bounds for T* is estimated as follows:
(i) If b < 0, then
(ii) If b = 0, then
(iii) If b > 0, then
or
3 Existence of solutions
In this section, we are going to obtain the existence of weak solutions to the problem (1.1) using FaedoGalerkin's approximation.
Theorem 1 Let the assumptions (A1)(A3) hold. Then there exists at least a solution u of (1.1) satisfying
and
as t → 0.
Proof We choose a basis {ω_{k}} (k = 1, 2, ...) in which is orthonormal in L^{2}(Ω) and ω_{k }being the eigenfunctions of biharmonic operator subject to the homogeneous Dirichlet boundary condition.
Let V_{m }be the subspace of generated by the first m vectors. Define
where u_{m}(t) is the solution of the following Cauchy problem
with the initial conditions (when m → ∞)
The approximate systems (3.3) and (3.4) are the normal one of differential equations which has a solution in [0, T_{m}) for some T_{m }> 0. The solution can be extended to the [0, T] for any given T > 0 by the first estimate below.
First estimation. Substituting instead of ω_{k }in (3.3), we find
Simple calculation similar to [11] yield
Combining (3.5) and (3.6), we find
integrating (3.7) over (0, t) and using assumption (A3) we infer that
where C_{1 }is a positive constant depending only on ∥u_{0}∥, ∥u_{1}∥, p, and l. It follows from (3.8) that
Second estimation. Differentiating (3.3) with respect to t, we get
If we substitute instead of ω_{k }in (3.10), it holds that
Since H^{2}(Ω) ↪ L^{2p+2}(Ω), using Lemma 2, Hölder and Young's inequalities and (3.8)
Combining the relations (3.11), (3.12) and integrating over (0, t) for all t ∈ [0, T] with arbitrary fixed T, we obtain
From (3.4) and (3.8), we deduce that
where L_{2 }is a positive constant independent of m. In the following, we find the upper bound for . Again we substitute instead of ω_{k }in (3.3), and choosing t = 0, we arrive at
which combined with the Green's formula imply
By using (A1), (3.4) and Young's inequality, we deduce that
where L_{3 }> 0 is a constant independent of m.
Owing to (3.8), (3.5) and Young's inequality with (A3), we deduce that
and
Now we choose γ > 0 small enough and combining (A3), (3.8), (3.13), (3.14), and (3.16)(3.20), we get
By using Gronwall's lemma we arrive at
for all t ∈ [0, T], and L_{10 }is a positive constant independent of m. Estimate (3.22) implies
By attention to (3.9) and (3.23), there exists a subsequence {u_{i}} of {u_{m}} and a function u such that
By AubinLions compactness lemma [18], it follows from (3.24) that
In the sequel we will deal with the nonlinear term. By (3.9) and Sobolev embedding theorem, we obtain
and therefore we can extract a subsequence {u_{i}} of {u_{m}} such that
Applying (3.24), (3.27) and letting i → ∞ in (3.3), we see that u satisfies the equation. For the initial conditions by using (3.4), (3.25) and the simple inequality
we get the first initial condition immediately. In the similar way, we can show the second initial condition and the proof is complete.
4 Blowup of solutions
In this section, we study blowup property of solutions with nonpositive initial energy as well as positive initial energy, and estimate the lifespan of solutions. For this purpose, we assume that g is positive and C^{1 }function satisfying
(A4)
and we make the following extra assumption on g
(A5)
From (2.1), (A4) and Lemma 1, we have
where . It is easy to verify that G(λ) has a maximum at and the maximum value is .
Lemma 4 Let (A4) hold andu be a local solution of (1.1). Then E(t) is a nonincreasing function on [0, T] and
for almost every t ∈ [0, T].
Proof Multiplying (1.1) by u_{t}, integrating over Ω, and finally integrating by parts, we obtain (4.2) for any regular solution. Then by density arguments, we have the result.
Lemma 5 Let (A4) hold and u be a local solution of (1.1) with initial data satisfying E(0) < E_{1 }and . Then there exists λ_{2 }> λ_{1 }such that
Proof See Li and Tsai [11].
The choice of the functional is standard (see [19])
It is clear that
and from (1.1)
Lemma 6 Let u be a solution of (1.1) and (A4), (A5) hold, then we have
Proof Using the Hölder and Young's inequalities, we arrive at
therefore (4.6) becomes
Then, using (4.2), we obtain
and so by (2.5) and (A5), we deduce
if we set then inequality (4.8) yields the desired result.
Consequently, we have the following result.
Lemma 7 Assume that (A4) and (A5) hold. u be a local solution of (1.1) and that either one of the following four conditions is satisfied:
(i) E(0) < 0
(ii) E(0) = 0 and ψ'(0) > 0
Then ψ' (t) > 0 for t > t*, where
in case (i)
in cases (ii), (iv)
and in case (iii)
Proof Suppose that condition (i) is satisfied. Then from (4.5), we have
Thus ψ'(t) > 0 for t > t*, and it is easy to see that t* satisfies (4.9).
If E(0) = 0, then by using (4.3) we have ψ" (t) ≥ 0, and since ψ'(0) > 0 we arrive at
If and then by Lemma 4, we see that
Thus from (4.5), we have
and integrating (4.12) from 0 to t gives
where t* satisfies (4.11).
Let , this assumption causes that
and by using Hölder and Young's inequalities, we get
thus
We see that the hypotheses of Lemma 2 are fulfilled with
and the conclusion of Lemma 2.2 gives us
Therefore the proof is complete.
To estimate the lifespan of ψ(t), we define the following functional
Then we have
Using (4.4)(4.6) and exploiting Holder's inequality on ψ'(t), we get
Utilizing the last inequality into (4.16) yields
Now we should assume different values for initial energy E(0).
(1) At first if E(0) ≤ 0 then from (4.17) we have
on the other hand by Lemma 7, Y'(t) < 0 for t > t*. Multiplying (4.18) by Y'(t) and integrating from t* to t, we deduce that
where
and
Then the hypotheses of Lemma 3 are fulfilled with and using the conclusion of Lemma 3, there exists a finite time T* such that , i.e., in this case some solutions blow up in finite time T*.
(2) If , then from (4.17) and (4.12) we have
Then using the same arguments as in (1), we get
where
and
Thus by Lemma 3, there exists a finite time T* such that
(3) . In this case, it is easy to see that by using (4.19) and (4.20) into discussion in part (1), we obtain
Hence, Lemma 3 yields the blowup property in this case.
Therefore, we proved the following theorem.
Theorem 2 Assume that (A4) and (A5) hold. u be a local solution of (1.1) and that either one of the following four conditions is satisfied:
(i) E(0) < 0
(ii) E(0) = 0 and ψ'(0) > 0
Then the solution u blows up at finite time T*. Moreover, the upper bounds for T* can be estimated according to the sign of E(0):
in case (i)
in cases (ii)
in case (iii)
and in case (iv)
where . Here α, β, α_{1}, and β_{1 }are given in (4.19)(4.22), respectively. Note that each t* in the above cases satisfy the same case in Lemma 7.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for the careful reading of this article and for the valuable suggestions to improve the presentation and style of the article. This study was supported by Shiraz University.
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