Life span of nonnegative solutions to an evolution equation

Abstract

We study the porous medium equation with a fractional integral source. The model is considered in the whole Euclidean space and is supplemented with a nonnegative, bounded and nontrivial initial condition. We show that there exists a critical exponent separating two different behaviors of solutions.When the nonlinearity is below this critical value, the problem admits no global nontrivial solution. This fact allows us to derive an upper bound for the maximal time of existence of solutions. In particular, we prove that solutions exist only on finite time interval and provide an estimate for the lifespan in terms of the size of initial data.