A general blow-up result on nonlinear boundary-value problems on exterior domains

Abstract

In the¯rst part, we study several exterior boundary-value problems covering three types of semilinear equations: elliptic, parabolic and hyperbolic. By a uni¯ed approach, we show that these problems share a common critical behaviour. In the second part we prove a blow-up result for an inhomogeneous porous medium equation with the critical exponent, which was left open in a previous paper.

The Fujita exponent for semilinear heat equations with quadratically decaying potential or in an exterior domain

“…15), we conclude that (3.10) holds with the integer N 0 replaced by nonintegral N . From this and (3.9) we then also obtain (3.4) with the integer N 0 replaced by non-integral N .…”

“…In the case a ≡ 1, n ≥ 3 and p = p * , the result in Theorem 4 was proven in [2]. For some other works that treat the critical exponent in exterior domains, see [9] and [15]. Most of the results in these papers do not cover the case in which p is equal to the critical exponent.…”

Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains

“…A natural question is to understand the wave equation or inequality on other unbounded domains of R N . The study of blow-up for wave equation on exterior domains was initialized by Zhang in [18]. Among many other things, he considered the inhomogeneous equation 9) where N ≥ 3, α > −2 and Ω ⊂ R N is a smooth bounded set.…”

“…In this work, we make use of harmonic function on Ω c with zero boundary condition, which permits to cut off only at infinity. These ideas make our method more transparent, for example we avoid the iterative step used in [18,16].…”

Blow-up theorems of Fujita type for a semilinear parabolic equation with a gradient term

“…There have been many kinds of extensions of Fujita's results since then, such as different types of parabolic equations and systems with or without degeneracies or singularities, various geometries of domains, different nonlinear reactions or nonhomogeneous boundary sources, etc. One can see the survey papers [2,3] and the references therein, and more recent work [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. For the Cauchy problem of…”