一类具超临界源的非线性黏弹性双曲方程解的爆破时间下界估计
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摘要
考虑一类涉及超临界源项的非线性黏弹性双曲方程解的爆破时间下界估计,通过构造带阻尼项的控制函数,利用能量估计方法和Sobolev嵌入定理,得到了该问题解爆破时间的显式下界估计.
We considered estimate of a lower bound of blow-up time for solutions to a class of nonlinear viscoelastic hyperbolic equations involving supercritical source terms.By constructing the control function with damping term,using the mehtod of energy estimate and Sobolev embedding theorem,we obtained an explicit estimate of lower bound of the blow-up time for the solutions to the problem.
We considered estimate of a lower bound of blow-up time for solutions to a class of nonlinear viscoelastic hyperbolic equations involving supercritical source terms.By constructing the control function with damping term,using the mehtod of energy estimate and Sobolev embedding theorem,we obtained an explicit estimate of lower bound of the blow-up time for the solutions to the problem.
引文
[1]COLEMAN B D,NOLL W.Foundations of Linear Viscoelasticity[J].Rev Modern Phys,1961,33:239-249.
[2]FABRIZIO M,MORRO A.Mathematical Problems in Linear Viscoelasticity[M]//SIAM Studies in Applied Mathematics,Vol.12.Philadelphia,PA:Society for Industrial and Applied Mathematics,1992.
[3]RENARDY M,HRUSA W J,NOHEL J A.Mathematical Problems in Viscoelasticity[M]//Pitman Monographs&Surveys in Pure&Applied Mathematics,Vol.35.New York:Longman Higher Education,1987.
[4]GUO Bin,GAO Wenjie.Blow-Up of Solutions to Quasilinear Hyperbolic Equations with p(x,t)-Laplacian Operator and Positive Initial Energy[J].Comptes Rendus Mécanique,2014,342:513-519.
[5]SUN Lili,GUO Bin,GAO Wenjie.A Lower Bound for the Blow-Up Time to a Damped Semilinear Wave Equation[J].Appl Math Lett,2014,37:22-25.
[6]GUO Bin,LIU Fang.A Lower Bound for the Blow-Up Time to a Viscoelastic Hyperbolic Equation with Nonlinear Sources[J].Appl Math Lett,2016,60:115-119.
[7]ZHOU Jun.Lower Bounds for Blow-Up Time of Two Nonlinear Wave Equations[J].Appl Math Lett,2015,45:64-68.
[8]GUO Bin.An Inverse H9lder Inequality and Its Application in Lower Bound Estimates for Blow-Up Time[J].Comptes Rendus Mécanique,2017,345(6):370-377.
[9]GUO Yanqiu,Rammaha M A,Sakuntasathien S.Blow-Up of a Hyperbolic Equation of Viscoelasticity with Supercritical Nonlinearities[J].J Differential Equations,2017,262(3):1956-1979.
[2]FABRIZIO M,MORRO A.Mathematical Problems in Linear Viscoelasticity[M]//SIAM Studies in Applied Mathematics,Vol.12.Philadelphia,PA:Society for Industrial and Applied Mathematics,1992.
[3]RENARDY M,HRUSA W J,NOHEL J A.Mathematical Problems in Viscoelasticity[M]//Pitman Monographs&Surveys in Pure&Applied Mathematics,Vol.35.New York:Longman Higher Education,1987.
[4]GUO Bin,GAO Wenjie.Blow-Up of Solutions to Quasilinear Hyperbolic Equations with p(x,t)-Laplacian Operator and Positive Initial Energy[J].Comptes Rendus Mécanique,2014,342:513-519.
[5]SUN Lili,GUO Bin,GAO Wenjie.A Lower Bound for the Blow-Up Time to a Damped Semilinear Wave Equation[J].Appl Math Lett,2014,37:22-25.
[6]GUO Bin,LIU Fang.A Lower Bound for the Blow-Up Time to a Viscoelastic Hyperbolic Equation with Nonlinear Sources[J].Appl Math Lett,2016,60:115-119.
[7]ZHOU Jun.Lower Bounds for Blow-Up Time of Two Nonlinear Wave Equations[J].Appl Math Lett,2015,45:64-68.
[8]GUO Bin.An Inverse H9lder Inequality and Its Application in Lower Bound Estimates for Blow-Up Time[J].Comptes Rendus Mécanique,2017,345(6):370-377.
[9]GUO Yanqiu,Rammaha M A,Sakuntasathien S.Blow-Up of a Hyperbolic Equation of Viscoelasticity with Supercritical Nonlinearities[J].J Differential Equations,2017,262(3):1956-1979.