Optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a Hardy potential

Abstract

We deal with the following parabolic problem{u(t) - Delta u = vertical bar del u vertical bar(p) + lambda u/vertical bar x vertical bar(2) + f, u > 0 in Omega x (0, T), u(x,t) = 0 on partial derivative Omega x (0, T), u(x,0) = u(0)(x), x is an element of Omega,where Omega subset of R-N, N >= 3, is a bounded regular domain such that 0 is an element of Omega OR Omega = R-N, p > 1, lambda >= 0 and f >= 0, u(0) >= 0 are in a suitable class of functions.There are deep differences with respect to the heat equation (lambda = 0). The main features in the paper are the following.If lambda > 0, there exists a critical exponent p(+)(lambda) such that for p >= p(+)(lambda), there is no nontrivial local solution.p(+)(lambda) is optimal in the sense that, if p < p(+)(lambda) there exists solution for suitable data.If we consider the Cauchy problem, i.e., Omega equivalent to R-N, we find the same phenomenon about the critical power p(+)(lambda) as above. Moreover, there exists a Fujita type exponent F(lambda) < p(+)(lambda) in the sense that independently of the initial datum, for 1 < p < F(lambda), any solution blows up in a finite time respect to an integral norm. This is a major difference with respect to the heat equation (lambda = 0).