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Interface development for the nonlinear degenerate multidimensional reaction–diffusion equations

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Abstract

This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE

$$\begin{aligned} u_t-\Delta u^m +b u^\beta =0, \quad x\in {\mathbb {R}}^N, 0<t<T \end{aligned}$$

with nonnegative initial function \(u_0\) such that

$$\begin{aligned} supp~u_0 = \{|x|<R\}, \ u_0 \sim C(R-|x|)^\alpha ,~{as} \ |x|\rightarrow R-0, \end{aligned}$$

where \(m>1, C,\alpha , \beta >0, b \in {\mathbb {R}}\). Interface surface \(t=\eta (x)\) may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction terms near the boundary of support, expressed in terms of the parameters \(m,\beta , \alpha , sign\ b\) and C. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface.

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Appendix

Appendix

Here we bring explicit values of the constants used in Sects. 2 and 5 .

$$\begin{aligned} \zeta _1= & {} {\left\{ \begin{array}{ll} -A_1^{\frac{m-1}{2}}\big (1+b(1-\beta )A_1^{\beta -1}\big )^{-\frac{1}{2}} \big (2m(m+\beta )(1-\beta )\big )^{\frac{1}{2}}(m-\beta )^{-1}&{}~\quad \text {if} \ m+\beta>2\\ -(A_1/C_*)^{\frac{m-\beta }{2}}&{}~\quad \text {if} \ 1\le m+\beta<2, \end{array}\right. }\\ C_1= & {} {\left\{ \begin{array}{ll} A_1(-\zeta _1)^{\frac{2}{m-\beta }}&{}~\quad \text {if} \ m+\beta>2\\ C_*&{}~\quad \text {if} \ 1\le m+\beta<2, \end{array}\right. }\\ A_1= & {} w(0,1)=h(0).\\ \zeta _2= & {} {\left\{ \begin{array}{ll} C^{\frac{\beta -m}{2}}\big (b(1-\beta )\big (1-\big (C/C_*\big )^{m-\beta } \big )\big )^{\frac{m-\beta }{2(1-\beta )}} \quad \text {if}~ m+\beta >2\\ \delta _*\Gamma l_1\quad \text {if}~ m+\beta <2, \end{array}\right. }\\ l_1= & {} C^{\frac{\beta -m}{2}}\Big [b(1-\beta )\Big (\delta _*\Gamma \Big )^{-1} \Big (1-\delta _*\Gamma -\Big (1-\delta _*\Gamma \Big )^{-1} \big (C/C_*\big )^{m-\beta }\Big )\Big ]^{\frac{m-\beta }{2(1-\beta )}},\\ \Gamma= & {} 1-\big (C/C_*\big )^{\frac{m-\beta }{2}},\quad C_2=C \big (1-\delta _*\Gamma \big )^{\frac{2}{\beta -m}},\\&\quad \text {and } \delta _*\in (0,1) \text { satisfies}\\ g(\delta _*)= & {} \max _{[0,1]}g(\delta ),\quad g(\delta )={\delta }^{\frac{2-\beta -m}{m-\beta }}\Big [1-\delta \Gamma -\Big (1-\delta \Gamma \Big )^{-1}\big (C/C_*\big )^{m-\beta }\Big ] \end{aligned}$$

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Abdulla, U.G., Abuweden, A. Interface development for the nonlinear degenerate multidimensional reaction–diffusion equations. Nonlinear Differ. Equ. Appl. 27, 3 (2020). https://doi.org/10.1007/s00030-019-0606-2

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