Blow-up rates for semi-linear reaction-diffusion systems

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• 作者：Henghui Zou
• 关键词：35K51 ; 35K57 ; 35K58
• 刊名：Journal of Differential Equations
• 出版年：1 August 2014
• 年：2014
• 卷：257
• 期：3
• 页码：843-867
• 全文大小：323 K

文摘

Let 惟⊂Rⁿ$惟\subset {\mathbb{R}}^n$ (n≥1$n\ge 1$) be a bounded smooth domain. Consider the following initial–boundary value problem of reaction–diffusion systems where u=(u₁,u₂)≥0$\mathbf{u}=(u_1,u_2)\ge 0$, and 桅(x)=(蠁₁(x),蠁₂(x))≥0$\mathit{桅}(x)=({蠁}_1(x),{蠁}_2(x))\ge 0$, and 谓   is the unit outer normal at ∂惟   and T∈(0,∞]$T\in (0,\infty ]$ is the maximum existence time of u (in L^∞$L^{\infty }$-norm) and the exponents p_ij$p_{ij}$, i,j=1,2$i,j=1,2$, are non-negative real numbers.

Systems of form (I) naturally arise in studying non-linear phenomena in biology, chemistry, medicine and physics. For instance, (I) has been used to model densities and temperatures in chemical reactions, condensate amplitudes in Bose–Einstein condensates, wave amplitudes (or envelops of multiple interacting optical modes) in optical fibers, and pattern formation in ecological systems.

Under suitable conditions on p_ij$p_{ij}$, i,j=1,2$i,j=1,2$, we established the following exact   blow-up rates for blow-up solutions u of (I)

where 胃₁${胃}_1$ and 胃₂>0${胃}_2>0$ are positive exponents depending only on p_ij$p_{ij}$, generalizing earlier results in this direction.