Displaying similar documents to “On critical exponents for a system of heat equations coupled in the boundary conditions.”

Nonlinear Heat Equation with a Fractional Laplacian in a Disk

Vladimir Varlamov (1999)

Colloquium Mathematicae

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For the nonlinear heat equation with a fractional Laplacian u t + ( - Δ ) α / 2 u = u 2 , 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution...

On the Kuramoto-Sivashinsky equation in a disk

Vladimir Varlamov (2000)

Annales Polonici Mathematici

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We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed...

The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation

Juan Luis Vázquez (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.