高维空间上非线性抛物方程在非线性边界条件下解的爆破时间的下界

Lower Bound for the Blow-up Time of a Nonlinear Parabolic Equation under Nonlinear Boundary Conditions in High Dimensional Spaces

研究了高维空间上非线性抛物方程在非线性边界条件下的解的爆破问题。通过构造一个能量表达式,运用微分不等式的方法,得到该能量方程所满足的微分不等式。然后通过积分得到了高维空间上当爆破发生时解在非线性边界条件下的爆破时间的下界。

and Applied Analysis,2014:1-9.[7]LIU Y,LUO S G,YE Y H.Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions[J].Computers and Mathematics with Applications,2013,65(8):1194-1199.[8]LIU Z Q,FANG Z B.Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux[J].Discrete and Continuous Dynamical Systems-Series B,2016,21(10):3619-3635.[9]SHEN X H,DING J T.Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions[J].Computers and Mathematics with Applications,2019,77(12):3250-3263.[10]LIU Y.Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions[J].Mathematical and Computer Modelling,2013,57(3/4):926-931.[11]CHEN W H,LIU Y.Lower bound for the blow up time for some nonlinear parabolic equations[J].Boundary Value Problems,2016(1):1-6.[12]TANG G S.Blow-up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions[J].Computers and Mathematics with Applications,2017,74(3):360-368.[13]BREZIS H.Functional analysis,sobolev spaces and partial differential equations,in:Universitext[M].New York:Springer,2011:280-284.Lower Bound for the Blow-up Time of a Nonlinear Parabolic Equation under Nonlinear Boundary Conditions in High Dimensional SpacesOUYANG Bai-ping,LI Yuan-fei(College of Data Science,Huashang College Guangdong University of Finance&Economics,Guangzhou 511300,China)Abstract:The paper deals with the blow up phenomenon for the nonlinear parabolic equation under nonlinear boundary conditions in high dimensional spaces.An energy expression is constructed first and then a differential inequality that the energy expression satisfied is derived by using the technique of a differential inequality.Integrating the inequality,the lower bound for the blow up time of the solution under nonlinear boundary conditions is obtained when blow-up does really occur in high dimensional spaces.