On the asymptotic behavior of solutions of certain quasilinear parabolic equations

Ni, Wei Ming; Sacks, Paul; Tavantzis, John

doi:10.1016/0022-0396(84)90145-1

Cited by 141 publications

(103 citation statements)

References 13 publications

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“…with Dirichlet boundary conditions, the question of boundedness of global solutions has been the subject of intensive investigation (see [2], [6]- [8], [12], [16] and the references therein, and [5,17] for recent surveys). In particular, unlike for (1.1), infinite time blow-up never occurs for (1.3) in space dimension 1 (or 2, but it can occur in 3 or more dimensions).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Infinite time blow-up for superlinear parabolic problems with localized reaction

Souplet

2004

Proc. Amer. Math. Soc.

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Abstract. We consider the nonlocal diffusion equationon the space interval (0, 1), with Dirichlet boundary conditions. It is known that if the curve x 0 (t) remains in a compact subset of (0, 1) for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on x 0 is sharp: for a large class of functions x 0 (t) approaching the boundary as t → ∞, blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Infinite time blow-up for superlinear parabolic problems with localized reaction

Souplet

2004

Proc. Amer. Math. Soc.

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“…We deduce from [22] and an obvious modification of [20] that there exists α * > 0 with the following property: if α < α * then w α (t) exists for all t ∈ R + and w α (t) → 0 as t → ∞; if α > α * then this solution blows up in finite time completely.…”

Section: Optimality Of the Growth Boundsmentioning

confidence: 91%

Optimal Control Problems with Final Observation Governed by Explosive Parabolic Equations

Amann

Quíttner²

2005

SIAM J. Control Optim.

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Abstract. We study optimal controls problems with final observation. The governing parabolic equations or systems involve superlinear nonlinearities and their solutions may blow up in finite time. Our proof of the existence, regularity and optimality conditions for an optimal pair is based on uniform a priori estimates for the approximating solutions. Our conditions on the growth of the nonlinearity are essentially optimal. In particular, we also solve a longstanding open problem of J.L. Lions concerning singular systems.

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“…[10]. In addition by standard arguments u(-,(;u o )-»w() uniformly, see [13] and the references therein. The hypotheses (Hi) implies that there exists a sequence of times (t n ) such that sup "*(".…”

Section: Convergence Of Unbounded Solutionsmentioning

confidence: 99%

“…As the next step we have the proposition: Considering the ball B(x o ,r o ), and using (3.7) we obtain Following Ni et al [13], a result of Gidas et al [5] may be applied for a convex region to give…”

Section: Blow-up For Large Initial Datamentioning

confidence: 99%

Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation

Tzanetis¹

1996

Proceedings of the Edinburgh Mathematical Society

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The initial-boundary value problem for the nonlinear heat equation u, = Au + Xf(u) might possibly have global classical unbounded solutions, u* = u(x, r; uS), for some "critical" initial data u*. The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x,A) for some values of L We find, for radial symmetric solutions, that u*(r,t)-*w{r) for any 0 < r < l but supu*(-,() = "*(0,t)-*°o> as t-»oo. Furthermore, if u 0 > u*, where u* is some such critical initial data, then u = u(x, t; u 0 ) blows up in finite time provided that / grows sufficiently fast.

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On the asymptotic behavior of solutions of certain quasilinear parabolic equations

Cited by 141 publications

References 13 publications

Infinite time blow-up for superlinear parabolic problems with localized reaction

Infinite time blow-up for superlinear parabolic problems with localized reaction

Optimal Control Problems with Final Observation Governed by Explosive Parabolic Equations

Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation

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