Generalized solutions of nonlinear parabolic equations with distributions as initial conditions

Abstract

We consider the nonlinear parabolic equation ut−Δu + u3 = 0 in Q = Ω×]0, T[ (T > 0, Ω open set in Rd, d = 1, 2, …) with the boundary condition u(x, t) = 0 on ∂Ω × ]0, T[ and the initial condition u(x, 0) = δ(x) in Ω, where δ is the Dirac mass at the origin of R. It is known that this problem has no weak solution in any known classical sense (within the distribution theory). Using a theory of generalized functions we obtain existence, uniqueness, and consistence results, which describe mathematically the behaviour of the solutions uε, obtained with smooth initial conditions uε(x, 0) = δε(x), δε ∈D(Ω), and δε → δ when ε → 0.