Existence and uniqueness of solutions of diffusion-absorption equations with general data


We study the nonlinear parabolic equation ut = Δ(um) = upin the exponent range 1 < m < p, with general initial data u(x, 0) = u0(x)∈RN. The emphasis is on the fact that in this exponent range we do not need any growth conditions on the data as x → ∞. We show existence and uniqueness of a strong nonnegative solution of the Cauchy problem with data u0 ∈ L1loc(RN), u0 ≥ 0. This solution is continuous and bounded fort ≥ T > 0. We discuss several extensions, like existence of solutions with changing sign, and existence of solutions when the initial datum is a locally finite measure. We also consider the problem posed in an open subset of RN with infinite boundary data. Finally, we comment on extensions to other related equations of a more general form. © 1994, Khayyam Publishing.

Publication Title

Differential and Integral Equations

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