Large-time behavior of solutions to the Cauchy problem for degenerate parabolic system

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.

Keywords:

• Degenerate parabolic systems
• large-time behavior
• optimal bound
• Extinction in finite time
• finite speed of propagation

AMS 2000 Mathematics Subject Classifications:

• 35B33
• 35B45
• 35K55
• 35K65

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The publication has been prepared with the support of the ‘RUDN University Program 5-100’ and funded by RFBR according to the research projects no. 12-34-56789 and no. 12-34-56789