Reaction–Diffusion in Irregular Domains

We consider the Cauchy–Dirichlet and Dirichlet problems for the nonlinear parabolic equationut−a(um)xx+buβ=0,where a>0, b R1, m>0, and β>0. The problems are considered in noncylindrical domains with nonsmooth boundaries. Existence, uniqueness, and comparison results are established. Constructed solutions are continuous up to the nonsmooth boundary if at each interior point the left modulus of the lower (respectively upper) semicontinuity of the left (respectively right) boundary curve satisfies an upper (respectively lower) Hölder condition near zero with Hölder exponent ν> . The value is critical as in the classical theory of the heat equation ut=uxx.