General decay rate estimates for a semilinear parabolic equation with memory term and mixed boundary condition
© Li et al.; licensee Springer 2014
Received: 24 February 2014
Accepted: 1 August 2014
Published: 26 September 2014
This work is concerned with a mixed boundary value problem for a semilinear parabolic equation with a memory term. Under suitable conditions, we prove that the energy functional decays to zero as the time tends to infinity by the method of perturbation energy, in which the usual exponential and polynomial decay results are only special cases.
where () is a bounded domain with sufficiently smooth boundary ∂ Ω such that , , and and have positive measures, ν is the unit outward normal vector on ∂ Ω, and g, a, f, and are memory kernel, coefficient, nonlinear, and initial functions, respectively, satisfying appropriate conditions; see (H1)-(H4) in Section 2.
where u is the temperature, the heat flux, λ the diffusion coefficient and the integral term represents the memory effect in the material. The memory kernel g is defined on and represents the negative derivative of the relation function of heat flux. The heat balance equation implies that will satisfy (1.1), provided that the temperature is assumed to be known for . The study on this type of equations has drawn considerable attention; see –. From the mathematical point of view, one would expect the integral term in the equation above is dominated by the leading term . Hence, the theory of parabolic equations can be applied to this type of equations.
To motivate our work, let us recall some results on the global existence, blow-up solutions, and asymptotic properties of the initial boundary value problems for semilinear parabolic equations and systems with or without memory term. In the absence of the memory term (), there are many results on the global existence and finite time blow-up of the solutions for the semilinear parabolic equation; see the monographs ,  and the survey papers –. Roughly summary, the global and nonglobal existences and the behavior of solutions depend on nonlinearity, dimension, initial data, and nonlinear boundary flux. Concerning systems, we refer to .
Moreover, there have also been published many other results for single equations with memory. We refer the readers to ,  and the references therein. Concerning systems, similar examples exist in the works of Pao  and Yamada , as well as others.
then the solution with small initial energy decays exponentially for and polynomially for . Thereafter, Messaoudi and Tellab  established a general decay result from which the usual exponential and polynomial decay results are only special cases.
Accessing the relevant papers, one can find that research on the asymptotic behavior of the solution for the nonlocal semilinear parabolic equation (1.1) with mixed boundary conditions has not been started yet. Very recently, for mixed boundary problem (1.1)-(1.4) with generalized Lewis functions, Fang and Qiu  proved the existence and uniqueness of global solution and the energy functional decays exponentially or polynomially to zero as the time tends to infinity by the technique of Lyapunov functional. Motivated by this observation, we intend to study the generalized property of energy decay for the initial mixed boundary value problem (1.1)-(1.4) using the technique of perturbation energy.
The rest of our paper is organized as follows: In Section 2, we present some assumptions, lemmas, and an energy functional, and give the energy decay results in Section 3.
Throughout this paper, We use the standard Lebesgue space , and the Sobolev space , with their usual scalar products and norms. To simplify the notations, we denote and by and , respectively.
We give the following general hypotheses on the memory kernel g, coefficient a, nonlinearity function f, and initial function :
where , , and .
where the set.
We recall the trace Sobolev embedding , and the embedding inequality , where is the optimal constant.
One can have the following nonincreasing property on .
for all and all .
By using a similar argument in , one can show the existence and uniqueness of the global solution to problem (1.1)-(1.4) with assumptions (H1)-(H4) by the technique of Galerkin, the contraction mapping principle, and a continuation argument.
3 General energy decay rate
Thus, selecting , we get (3.3).
We now give precise estimates of the derivatives and which will be used in the proof of our main results.
Proof of Lemma 2
By taking in the inequality above, we obtain (3.4), which completes the proof. □
Proof of Lemma 3
By taking in the inequality above, we get (3.6), which completes the proof. □
Proof of Theorem 1
where c is a positive constant. This completes the proof. □
It can be seen that the estimate (3.15) is also true for by the continuity and boundedness of and .
This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to sincerely thank all the reviewers for their insightful and constructive comments.
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