ABSTRACT

We consider nonnegative solutions to the Cauchy problem for a degenerate parabolic system of the form $\begin{array}{rl}{u}_t& =\mathrm{div}\left(v^{{\alpha }_1}\mathrm{\nabla }{u}^{m_1}\right),(x,t)\in S_T={\mathbf{R}}^N\times (0,T)\\ v_t& =\mathrm{div}\left(u^{{\alpha }_2}\mathrm{\nabla }{v}^{m_2}\right),(x,t)\in S_T\\ u(x,0)& =u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0,x\in {\mathbf{R}}^N,N\ge 1.\end{array}$ Under the suitable assumptions on the parameters of nonlinearities and initial data, we obtained optimal decay estimates of a solution for a large time. Moreover, the phenomena of extinction in finite time was established. Provided that initial data are compactly supported, we proved the property of finite speed of propagation.

中文翻译：

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退化抛物线系统柯西问题解的大时行为

摘要

我们考虑形式为退化抛物线系统的柯西问题的非负解$\begin{array}{rl}{你}_{吨}& =\mathrm{div}\left(v^{{\alpha }_1}\mathrm{\nabla }{你}^{{米}_1}\right),(X,吨)\in {\mathrm{小号}}_{吨}={\mathbf{R}}^{ñ}\times (0,吨)\\ v_{吨}& =\mathrm{div}\left({你}^{{\alpha }_2}\mathrm{\nabla }{v}^{{米}_2}\right),(X,吨)\in {\mathrm{小号}}_{吨}\\ 你(X,0)& ={你}_0\left(X\right)\ge 0,v(X,0)=v_0\left(X\right)\ge 0,X\in {\mathbf{R}}^{ñ},ñ\ge 1.\end{array}$在非线性参数和初始数据的适当假设下，我们得到了一个解的最优衰减估计。此外，还建立了有限时间内灭绝的现象。假设初始数据被紧凑地支持，我们证明了有限传播速度的性质。