具一般非线性项抛物型Kirchhoff方程解的有限时间爆破
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摘要
考虑一类具一般非线性项的抛物型Kirchhoff方程解的有限时间爆破问题,借助一阶微分不等式和凸方法,给出解在有限时刻爆破的一些充分条件,并得到了爆破时间的上界估计.
We considered the finite time blow-up of solutions to a class of parabolic type Kirchhoff equations with general nonlinearity.By using the first order differential inequality and convexity arguments,we gave some sufficient conditions for the solutions to blow up in finite time,and obtained the upper bound estimate of the blow-up time.
We considered the finite time blow-up of solutions to a class of parabolic type Kirchhoff equations with general nonlinearity.By using the first order differential inequality and convexity arguments,we gave some sufficient conditions for the solutions to blow up in finite time,and obtained the upper bound estimate of the blow-up time.
引文
[1]KIRCHHOFF G.Mechanik[M].Teubner:Leipzig,1883.
[2]LIONS J L.On Some Questions in Boundary Value Problems of Mathematical Physics[J].North-Holland Math Stud,1978,30:284-346.
[3]D’ANCONA P,SHIBATA Y.On Global Solvability of Non-linear Viscoelastic Equations in the Analytic Category[J].Math Methods Appl Sci,1994,17(6):477-486.
[4]D’ANCONA P,SPAGNOLO S.Global Solvability for the Degenerate Kirchhoff Equation with Real Analytic Data[J].Invent Math,1992,108(1):247-262.
[5]NISHIHARA K.On a Global Solution of Some Quasilinear Hyperbolic Equation[J].Tokyo J Math,1984,7(2):437-459.
[6]CHIPOT M,VALENTE V,CAFFARELLI G V.Remarks on a Nonlocal Problem Involving the Dirichlet Energy[J].Rend Sem Math Univ Padova,2003,110(4):199-220.
[7]ZHENG Songmu,CHIPOT M.Asymptotic Behavior of Solutions to Nonlinear Parabolic Equations with Nonlocal Terms[J].Asympt Anal,2005,45(3/4):301-312.
[8]HAN Yuzhu,LI Qingwei.Threshold Results for the Existence of Global and Blow-up Solutions to Kirchhoff Equations with Arbitrary Initial Energy[J].Comput Math Appl,2018,75(9):3283-3297.
[9]FU Yongqiang,XIANG Mingqi.Existence of Solutions for Parabolic Equations of Kirchhoff Type Involving Variable Exponent[J].Appl Anal,2016,95(3):524-544.
[10]LIAO Menglan,GAO Wenjie.Blow-up Phenomena for a Nonlocal p-Laplace Equation with Neumann Boundary Conditions[J].Arch Math,2017,108(3):313-324.
[11]PHILIPPIN G A,PROYTCHEVA V.Some Remarks on the Asymptotic Behaviour of the Solutions of a Class of Parabolic Problems[J].Math Methods Appl Sci,2006,29(3):297-307.
[12]LEVINE H A.Some Nonexistence and Instability Theorems for Solutions of Formally Parabolic Equations of the FormPutt=-Au+F(u)[J].Arch Rational Mech Anal,1973,51(5):371-386.
[13]SUN Fenglong,LIU Lishan,WU Yonghong.Finite Time Blow-Up for a Thin-Film Equation with Initial Data at Arbitrary Energy Level[J].J Math Anal Appl,2018,458(1):9-20.
[2]LIONS J L.On Some Questions in Boundary Value Problems of Mathematical Physics[J].North-Holland Math Stud,1978,30:284-346.
[3]D’ANCONA P,SHIBATA Y.On Global Solvability of Non-linear Viscoelastic Equations in the Analytic Category[J].Math Methods Appl Sci,1994,17(6):477-486.
[4]D’ANCONA P,SPAGNOLO S.Global Solvability for the Degenerate Kirchhoff Equation with Real Analytic Data[J].Invent Math,1992,108(1):247-262.
[5]NISHIHARA K.On a Global Solution of Some Quasilinear Hyperbolic Equation[J].Tokyo J Math,1984,7(2):437-459.
[6]CHIPOT M,VALENTE V,CAFFARELLI G V.Remarks on a Nonlocal Problem Involving the Dirichlet Energy[J].Rend Sem Math Univ Padova,2003,110(4):199-220.
[7]ZHENG Songmu,CHIPOT M.Asymptotic Behavior of Solutions to Nonlinear Parabolic Equations with Nonlocal Terms[J].Asympt Anal,2005,45(3/4):301-312.
[8]HAN Yuzhu,LI Qingwei.Threshold Results for the Existence of Global and Blow-up Solutions to Kirchhoff Equations with Arbitrary Initial Energy[J].Comput Math Appl,2018,75(9):3283-3297.
[9]FU Yongqiang,XIANG Mingqi.Existence of Solutions for Parabolic Equations of Kirchhoff Type Involving Variable Exponent[J].Appl Anal,2016,95(3):524-544.
[10]LIAO Menglan,GAO Wenjie.Blow-up Phenomena for a Nonlocal p-Laplace Equation with Neumann Boundary Conditions[J].Arch Math,2017,108(3):313-324.
[11]PHILIPPIN G A,PROYTCHEVA V.Some Remarks on the Asymptotic Behaviour of the Solutions of a Class of Parabolic Problems[J].Math Methods Appl Sci,2006,29(3):297-307.
[12]LEVINE H A.Some Nonexistence and Instability Theorems for Solutions of Formally Parabolic Equations of the FormPutt=-Au+F(u)[J].Arch Rational Mech Anal,1973,51(5):371-386.
[13]SUN Fenglong,LIU Lishan,WU Yonghong.Finite Time Blow-Up for a Thin-Film Equation with Initial Data at Arbitrary Energy Level[J].J Math Anal Appl,2018,458(1):9-20.