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# Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition

• In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point case, the existence of a unique self-similar profile is proven. Moreover, by constructing a Lyapunov function, we prove the convergence of the solution to the unique self-similar solution as $t$ approaches the blow-up time.
Mathematics Subject Classification: Primary: 35K20, 35K55; Secondary: 34A12.

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