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General decay rate estimates for a semilinear parabolic equation with memory term and mixed boundary condition
Boundary Value Problems volume 2014, Article number: 197 (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.
Our main interest lies in the following semilinear heat equation with a memory term:
subject to mixed boundary and initial conditions
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.
Many natural phenomena in engineering and physical science have been formulated with the nonlocal equation (1.1) as a mathematical model. For example, in the study of heat conduction in materials with memory, the classical Fourier law of the heat flux is replaced by the following form:
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 .
When a memory term exists (), Olmstead et al. considered the non-Newtonian fluid equation
subject to homogeneous Dirichlet boundary condition, and discussed that the bifurcation behavior. In , Bellout studied the following equation:
with homogeneous Dirichlet boundary condition, where is a smooth function and . The author established the existence and the uniqueness of the local classical solution, and obtained some criteria for solutions to blow up in a finite time. Moreover, he obtained some results on the blow-up points under some suitable assumptions. In , Yamada investigated the stability properties of the global solutions of the following nonlocal Volterra diffusion equation:
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.
Recently, Messaoudi  studied the semilinear heat equation with a source term of power form and homogeneous Dirichlet boundary condition
where the relaxation function is a bounded -function and , and proved the existence of a blow-up solution with positive initial energy by the convexity method. Later, Fang and Sun  improved the results of , when is replaced with a fully nonlinear source term . In , Berrimi and Messaoudi considered the quasilinear parabolic system
subject to homogeneous Dirichlet boundary condition, and proved that if a bounded square matrix such that
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 :
(H1) is a nonincreasing differentiable function such that, and there exists a differentiable function ξ satisfying
where , , and .
(H2) is a nonnegative bounded function such thatand
(H3) The functionis Lipschitz continuous and satisfies
(H4) (Compatibility condition) The initial functionsatisfies
where the set.
There are many functions satisfying (H1) and (H2). Examples of such functions are
Multiplying (1.1) by , integrating the result over Ω, and using Green’s formula, we can get
where we apply the fact that
We recall the trace Sobolev embedding , and the embedding inequality , where is the optimal constant.
One can have the following nonincreasing property on .
The energy functional is nonnegative and satisfies
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
In this section, we establish the estimates of general uniform energy decay rates and introduce a perturbed energy functional
to show the uniform decay of the solution, where and are positive constants, and
We can choose small and , if needed, so that
Indeed, through a simple calculation, we deduce that
where is an embedding constant satisfying the Poincaré inequality, . Hence, we have
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
The second term in the right-hand side of (3.5) is
Then we can deduce
By taking in the inequality above, we obtain (3.4), which completes the proof. □
Proof of Lemma 3
Now, we estimate the five terms in the right-hand side of (3.7):
By taking in the inequality above, we get (3.6), which completes the proof. □
Proof of Theorem 1
Since is positive, we have
By (2.1), (H1), and Lemmas 2 and 3, we have
for all . By choosing and so that
Multiplying (3.13) by , one can see that
from (3.13) and (H2). Let and then . Hence, we arrive at
by (H1) and (H3), where λ is a positive constant. A simple integration leads to
Again, employing is equivalent to leads to,
where c is a positive constant. This completes the proof. □
The exponential and polynomial decay estimates are only particular cases of Theorem 1. We illustrate the energy decay rate:
then (3.15) gives the exponential decay estimate
then we obtain the polynomial decay estimate
with and , then holds for
Thus, (3.15) gives the estimate
with , and (or and ), then for
we obtain from (3.15)
It can be seen that the estimate (3.15) is also true for by the continuity and boundedness of and .
Prato GD, Iannelli M: Existence and regularity for a class of integro-differential equations of parabolic type. J. Math. Anal. Appl. 1985, 112(1):36-55. 10.1016/0022-247X(85)90275-6
Friedman A: The IMA Volumes in Mathematics and Its Applications. Springer, New York; 1992.
Nohel JA: Nonlinear Volterra equations for heat flow in materials with memory. In Integral and Functional Differential Equations. Edited by: Herdman TL, Stech HW, Rankin SM III. Dekker, New York; 1981:3-82.
Yin HM: On parabolic Volterra equations in several space dimensions. SIAM J. Math. Anal. 1991, 22(6):1723-1737. 10.1137/0522106
Yin HM: Weak and classical solutions of some nonlinear Volterra integro-differential equations. Commun. Partial Differ. Equ. 1992, 17: 1369-1385. 10.1080/03605309208820889
Quittner R, Souplet P: Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States. Birkhäuser, Basel; 2007.
Hu B: Blow up Theories for Semilinear Parabolic Equations. Springer, Berlin; 2011.
Levine HA: The role of critical exponents in blow-up theorems. SIAM Rev. 1990, 32: 262-288. 10.1137/1032046
Lopez Gomez J, Marquez V, Wolanski N: Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition. J. Differ. Equ. 1991, 92(2):384-401. 10.1016/0022-0396(91)90056-F
Rodriguez-Bernal A, Tajdine A: Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differ. Equ. 2001, 169: 332-372. 10.1006/jdeq.2000.3903
Pucci P, Serrin J: Asymptotic stability for nonlinear parabolic systems. In Energy Methods in Continuum Mechanics. Kluwer Academic, Dordrecht; 1996:66-74. (Oviedo, 1994) 10.1007/978-94-009-0337-1_7
Olmstead WE, Davis SH, Rosenblat S, Kath WL: Bifurcation with memory. SIAM J. Appl. Math. 1986, 46(2):171-188. 10.1137/0146013
Bellout H: Blow-up of solutions of parabolic equation with nonlinear memory. J. Differ. Equ. 1987, 70: 42-68. 10.1016/0022-0396(87)90168-9
Yamada Y: On a certain class of semilinear Volterra diffusion equations. J. Differ. Equ. 1982, 88: 443-457.
Cannon JR, Lin Y:A priori error estimates for finite-element methods for nonlinear diffusion equations with memory. SIAM J. Numer. Anal. 1990, 27(3):595-607. 10.1137/0727036
Yong J, Zhang X: Exact controllability of the heat equation with hyperbolic memory kernel. In Control Theory of Partial Differential Equations. Chapman Hall CRC, Boca Raton; 2005:387-401. 10.1201/9781420028317.ch22
Pao CV: Solution of a nonlinear integrodifferential system arising in nuclear reactor dynamics. J. Math. Anal. Appl. 1974, 48: 470-492. 10.1016/0022-247X(74)90171-1
Yamada Y: Asymptotic stability for some systems of semilinear Volterra diffusion equations. J. Differ. Equ. 1984, 52: 295-326. 10.1016/0022-0396(84)90165-7
Messaoudi SA: Blow-up of solutions of a semilinear heat equation with a visco-elastic term. Prog. Nonlinear Differ. Equ. Appl. 2005, 64: 351-356.
Fang ZB, Sun L: Blow up of solutions with positive initial energy for the nonlocal semilinear heat equation. J. Korean Soc. Ind. Appl. Math. 2012, 16(4):235-242. 10.12941/jksiam.2012.16.4.235
Berrimi S, Messaoudi SA: A decay result for a quasilinear parabolic system. Prog. Nonlinear Differ. Equ. Appl. 2005, 53: 43-50.
Messaoudi SA, Tellab B: A general decay result in a quasilinear parabolic system with viscoelastic term. Appl. Math. Lett. 2012, 25: 443-447. 10.1016/j.aml.2011.09.033
Messaoudi SA: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal. TMA 2008, 69: 2589-2598. 10.1016/j.na.2007.08.035
Cavalcanti MM, Oquendo HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 2003, 42: 1310-1324. 10.1137/S0363012902408010
Fang ZB, Qiu LR: Global existence and uniform energy decay rates for the semilinear parabolic equation with a memory term and mixed boundary condition. Abstr. Appl. Anal. 2013., 2013:
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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Li, C., Qiu, L. & Fang, Z.B. General decay rate estimates for a semilinear parabolic equation with memory term and mixed boundary condition. Bound Value Probl 2014, 197 (2014). https://doi.org/10.1186/s13661-014-0197-0
- Heat Flux
- Memory Term
- Homogeneous Dirichlet Boundary Condition
- Memory Kernel
- Uniform Decay