张娜,方钟波

• 1:中国海洋大学数学科学学院

摘要(Abstract)：

本文针对具有非线性源的非散度型扩散方程初边值问题研究解的爆破性质。利用辅助函数法和修正微分不等式技巧等，建立适当的条件并使问题的解在新测度意义下有限时刻发生爆破。同时，在全空间中给出爆破解的生命跨度估计值。

关键词(KeyWords)： 非散度型扩散方程;非线性源;爆破解

Abstract:

Keywords:

基金项目(Foundation): 山东省自然科学基金项目(ZR2019MA072)资助~~

作者(Author): 张娜,方钟波

DOI: 10.16441/j.cnki.hdxb.20240321

参考文献(References)：

• [1] Zhou W S,Wu Z Q.Some results on a class of degenerate parabolic equations not indevergence form[J].Nonlinear Analysis,2005,60:863-886.
• [2] 金春花，尹景学.具非线性源的非散度型扩散方程的临界指标[J].数学年刊A辑，2009,30A(4):525-538.Jin C H,Yin J X.Critical exponent of a doubly degenerate parabolic equation in non-divergence form with nonlinear sources[J].Chinese Journal of Contemporary Mathematics,2009,30A(4):525-538.
• [3] Allen L J S.Persistence and extinction in single-species reaction-diffusion models[J].Bulletin of Mathematical Biology,1983,45:209-227.
• [4] Low B C.Resistive diffusion of force-free magnetic fields in a passive medium[J].The Astrophysical Journal,1973,181:209-226.
• [5] Friedman A,McLeod B.Blow-up of solutions of nonlinear degenerate parabolic equations[J].Archive for Rational Mechanics and Analysis,1986,96(1):55-80.
• [6] Epstein C L,Weinstein M I.A stable manifold theorem for the curve shortening equation[J].Communications on Pure and Applied Mathematics,1987,40:119-139.
• [7] DiBenedetto E.Degenerate Parabolic Equations[M].New-York:Springer-Verlag,1993.
• [8] Samarskii A A,Galaktionov V A,Kurdyumov S P,et al.Blow-up in Quasilinear Parabolic Equations[M].Berlin:Walter de Gruyter,1995.
• [9] Wu Z Q,Zhao J N,Yin J X,et al.Nonlinear Diffusion Equations[M].Singapore:World Scientific,2001.
• [10] Quittner R,Souplet P.Superlinear Parabolic Problems:Blow-Up,Global Existence and Steady States[M].Basel:Birkh?user Advanced Texts,2019.
• [11] Wang S,Wang M X,Xie C H.A nonlinear degenerate diffusion equation not in divergence form[J].Zeitschrift für Angewandte Mathematik und Physik,2000,51:149-159.
• [12] Wiegner M.A critical exponent in a degenerate parabolic equation[J].Mathematical Methods in the Applied Sciences,2002,25(11):911-925.
• [13] Wang M X.Some degenerate and quasilinear parabolic systems not in divergence form[J].Journal of Mathematical Analysis and Applications,2002,274:424-436.
• [14] Winkler M.Blow-up of solutions to a degenerate parabolic equation not in divergence form[J].Journal Differential Equations,2003,192:445-474.
• [15] Wang M X,Xie C H.A degenerate and strongly coupled quasilinear parabolic system not in divergence form[J].Zeitschrift für Angewandte Mathematik und Physik,2004,55:741-755.
• [16] Passo R Dal,Luckhaus S.A degenerate diffusion problem not in divergence form[J].Journal of Differential Equations,1987,69:1-14.
• [17] Mizoguchi N.Periodic solutions for degenerate diffusion equations[J].Indiana University Mathematics Journal,1995,44(2):413-432.
• [18] Giga Y,Mizoguchi N.On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type[J].Journal of Mathematical Analysis and Applications,1996,201:396-416.
• [19] Wang C P,Yin J X.Traveling wave fronts of a degenerate parabolic equation with nondivergence form[J].Journal of Partial Differential Equations,2003,16:62-74.
• [20] Zhou W S,Yao Z A.Behaviors of solutions for a singular diffusion equation[J].Journal of Mathematical Analysis and Applications,2007,327(1):611-619.
• [21] Souplet P h.Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions[J].Differential and Integral Equations,2002,15(2):237-256.
• [22] Yi S C,Fang Z B.Blow-up phenomena for a reaction-diffusion equation with nonlocal gradient terms[J].Taiwanese Journal of Mathematics,2023,27(4):737-757.
• [23] Jan B,Tomasz C,Cristian M R.Global existence vs.blow-up in a fully parabolic quasilinear 1D Keller-Segel system[J].Nonlinear Analysis,2012,75(13):5215-5228.
• [24] Payne L E,Philippin G A.Blow-up phenomena for a class of parabolic systems with time dependent coefficients[J].Applied Mathematics,2012,3(4):325-330.
• [25] Talenti G.Best constants in sobolev inequality[J].Annali di Matematica Pura ed Applicata,1976,110:353-372.

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