Existence results for a class of porous medium type equations with a quadratic gradient term

Abstract

In this paper we study the Dirichlet problem in Q T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term: $$u_t-div(a(x, t, u,\nabla u)) = b(x, t, u,\nabla u) + f (x, t)$$ The principal part of the operator degenerates in u = 0 according to a nonnegative increasing real function α(u), and the term \(b(x, t, u, \nabla u)\) grows quadratically with respect to the gradient. We prove an existence result for solutions to this problem in the framework of the distributional solutions under the hypotheses that both f and the initial datum u 0 are bounded nonnegative functions. Moreover as further results we get an existence result for the model problem $$u_t-div(\alpha(u)Du) = \beta(u)|\nabla u|^2 + f$$ in the case that the principal part of the operator is of fast-diffusion type, i.e. α(u) = u m , with −1 < m < 0.