Stabilization of solutions of weakly singular quenching problems

Abstract

In this paper we prove that if 0 > β > 1 , D ⊂ R N 0 > \beta > 1,\;D \subset {R^N} is bounded, and λ > 0 \lambda > 0 , then every element of the ω \omega -limit set of weak solutions of \[ u t − Δ u + λ u − β χ u > 0 = 0 in D × [ 0 , ∞ ) , u = { 1 on ∂ D × ( 0 , ∞ ) , u 0 > 0 on D ¯ × { 0 } \begin {array}{*{20}{c}} {{u_t} - \Delta u + \lambda {u^{ - \beta }}{\chi _{u > 0}} = 0\quad {\text {in}}\;D \times [0,\infty ),} \\ {u = \left \{ \begin {gathered} 1\qquad \quad {\text {on}}\;\partial D \times (0,\infty ), \hfill \\ {u_0} > 0\quad {\text {on}}\;\bar D \times \{ 0\} \hfill \\ \end {gathered} \right .} \\ \end {array} \] is a weak stationary solution of this problem. A consequence of this is that if D D is a ball, λ \lambda is sufficiently small, and u 0 {u_0} is a radial, then the set { ( x , t ) | u = 0 } \{ (x,t)|u = 0\} is a bounded subset of D × [ 0 , ∞ ) D \times [0,\infty ) .