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Critical Fujita absorption exponent for evolution $$p$$-Laplacian with inner absorption and boundary flux. (English) Zbl 1340.35181
This paper studies the evolutionary equation with the $$p$$-Laplacian and inner absorption in one space dimension, i.e., $\frac {\partial u}{\partial t} = \frac {\partial}{\partial x} \left ( \left | \frac {\partial u}{\partial x} \right | ^{p-2} \frac {\partial u}{\partial x} \right) - \lambda \, u^{\kappa} \,,\qquad (x,t)\in Q = \mathbb {R}_+\times \mathbb {R}_+ \,,\tag{P$$_p$$}$ subject to the following nonlinear boundary flux condition and the initial value condition, respectively, \begin{alignedat}{2} -\left | \frac {\partial u}{\partial x} \right | ^{p-2} \frac {\partial u}{\partial x} & = u(0,t)^q \,,\qquad & t > 0 \,,\tag{P$$_{\text{bc}}$$} \\ u(x,0) &= u_0(x) \,,\qquad & x > 0 \,.\tag{P$$_{\text{ic}}$$} \end{alignedat} It is assumed that $$2 < p < \infty$$, the constants $$\kappa$$, $$q$$, and $$\lambda$$ are given positive numbers, and $$u_0\colon \mathbb {R}_+\to \mathbb {R}_+$$ is a continuous function with compact support.
The paper is concerned with the existence and nonexistence of global (in time) solutions as well as their localization and nonlocalization in space (compact or noncompact support in the space variable $$x$$). It is proved that the critical boundary flux exponent $$q$$ is given by the formula $q^{\ast} = \max \left \{ \frac {2(p-1)}{p} \,,\,\frac {(\kappa + 1)(p-1)}{p} \right \} \,.$ The critical case $$q = q^{\ast}$$ is treated separately. The restriction to the space dimension allows for explicit calculations with the use of ordinary differential equations in the proofs.

##### MSC:
 35K92 Quasilinear parabolic equations with $$p$$-Laplacian 35B33 Critical exponents in context of PDEs 35B44 Blow-up in context of PDEs