Taiwanese Journal of Mathematics

CRITICAL EXPONENT FOR A SYSTEM OF SLOW DIFFUSION EQUATIONS WITH BOTH REACTION AND ABSORPTION TERMS

Sheng-Chen Fu

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Abstract

Let $\Omega$ is a bounded domain in $R^N$ with a smooth boundary $\partial\Omega$, $m,n\gt 1$ and $p,q,r,s,a,b$ are positive constants. For the initial and boundary value problem $$ \begin{array}{rl} u_t =\bigtriangleup u^m+v^p-au^r,& \quad x\in\Omega,\quad t\gt 0,\\ v_t =\bigtriangleup v^n+u^q-bv^s, & \quad x\in\Omega,\quad t\gt 0,\\ u=v=0, & \quad x\in\partial\Omega,\quad t\gt 0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),& \quad x\in\Omega, \end{array} $$ we prove that all solutions are globally bounded if $pq \lt \max\{n, r\} \max\{n, s\}$; while there are finite time blowing up solutions if $pq \lt \max\{n, r\} \max\{n, s\}$ and the initial data are sufficiently large. For the critical case $pq=\max\{m,r\}$ $\max\{n,s\}$, the existence or nonexistence of global solutions depends on the relation between the exponents $m$, $n$, $r$, $s$, and also the range of the parameters $a$, $b$.

Article information

Source
Taiwanese J. Math., Volume 13, Number 1 (2009), 307-316.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405285

Digital Object Identifier
doi:10.11650/twjm/1500405285

Mathematical Reviews number (MathSciNet)
MR2489320

Zentralblatt MATH identifier
1178.35054

Subjects
Primary: 35K65: Degenerate parabolic equations 35B33: Critical exponents

Keywords
critical exponent degenerate parabolic system

Citation

Fu, Sheng-Chen. CRITICAL EXPONENT FOR A SYSTEM OF SLOW DIFFUSION EQUATIONS WITH BOTH REACTION AND ABSORPTION TERMS. Taiwanese J. Math. 13 (2009), no. 1, 307--316. doi:10.11650/twjm/1500405285. https://projecteuclid.org/euclid.twjm/1500405285


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