A one-dimensional nonlinear heat equation with a singular term

Abstract

In this paper we are concerned with the Dirichlet problem for the one-dimensional nonlinear heat equation with a singular term: { u t = u x x − σ u m u x 2 + f ( x , t ) , u > 0 , ( x , t ) ∈ Q T , u ( a , t ) = u ( b , t ) = 0 , t ∈ [ 0 , T ] , u ( x , 0 ) = u 0 ( x ) , x ∈ I , where T > 0 , Q T = I × ( 0 , T ] , I = ( a , b ) with a < b , σ > 0 , − 1 ⩾ m > − 2 . We find that the problem may have multiple weak solutions for some initial data. To prove this, we need to study existence of positive classical solutions. In addition, we also discuss existence of a positive stationary solution for the above problem and relations between solutions of the above problem and the following problem: { u t = u x x + f ( x , t ) , ( x , t ) ∈ Q T , u ( b , t ) = u ( a , t ) = 0 , t ∈ [ 0 , T ] , u ( x , 0 ) = u 0 ( x ) , x ∈ I .