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Zbl 0936.35034
Mukai, Kentaro; Mochizuki, Kiyoshi; Huang, Qing
Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values. (English)
[J] Nonlinear Anal., Theory Methods Appl. 39, No.1, A, 33-45 (2000). ISSN 0362-546X

The following initial value problem is considered $$\partial_t u=\Delta u^m+u^p,\quad x\in \bbfR^N,\ t>0 \qquad u(x,0)=u_0(x),\quad x\in\bbfR^N,$$ where $1<m<p$, $N\ge 1$, and $u_0(x)$ is a nonnegative bounded and continuous function. The problem describes a combustion process in a stationary medium, where $u$ represents the temperature, and it is assumed that thermal conductivity and volume heat source depend on some powers of $u$. It is well known that this problem has a unique, nonnegative and bounded solution in some weak sense at least locally in time. The paper establishes some sufficient conditions implying that the considered solution exists only on a finite time interval (blow up) or that it has an infinite life span. It concentrates on the case of initial values $u_0$ having slow decay $u_0\sim\lambda|x|^a$, $\lambda>0$, $a\ge 0$, near $x=\infty$. The problems of global existence and nonexistence, large time behavior or life span are investigated in terms of $\lambda$ and $a$.
[Ivan Ginchev (Varna)]

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
35K65 Parabolic equations of degenerate type
35K15 Second order parabolic equations, initial value problems
35K55 Nonlinear parabolic equations
80A25 Combustion, interior ballistics

Keywords: blow up; global existence

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Zentralblatt MATH Berlin [Germany]