Blow-Up for Quasilinear Heat Equations Described by Means of Nonlinear Hamilton–Jacobi Equations

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It is well-known that the nonnegative solutions of the semilinear heat equation[formula]blow up in a finite timeT(depending on the initial data, assumed to be large enough). This equation is interesting because it exhibits in differentβ-ranges the three most typical blow-up behaviours for solutions of nonlinear parabolic equations. Indeed, we consider radialy symmetric solutions and show that forβ>2 single-point blow-up occurs, forβ<2 blow-up is global, and forβ=2 we have regional blow-up. Moreover, the analysis shows that the precise asymptotic behaviour is described by a nonconstant self-similar blow-up solution of the first-order Hamilton–Jacobi equation[formula]This means that both equations are asymptotically equivalent near blow-up. This type of asymptotic “degeneracy” of a parabolic equation into a first-order equation is actually proved for a more general class of quasilinear heat equations.