GLOBAL EXISTENCE, EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS

doi:10.1007/s10473-019-0508-8

Abstract

Abstract: In this paper, we study the initial-boundary value problem for the semilinear parabolic equations u_t -△_Xu=|u|^p-1u, where X=(X₁, X₂, …, X_m) is a system of real smooth vector fields which satisfy the H?rmander's condition, and is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.

Key words: finitely degenerate parabolic equation, global existence, blow-up, decay estimate

CLC Number:

35K58

Hua CHEN, Huiyang XU. GLOBAL EXISTENCE, EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS[J].Acta mathematica scientia,Series B, 2019, 39(5): 1290-1308.

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