Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time

Levine, Howard A.; Payne, L. E.

doi:10.1016/0022-0396(74)90018-7

Cited by 214 publications

(96 citation statements)

References 6 publications

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“…See for example [2], [12], [13], [14], [15], [21], [22], [29], [34]. We also refer to the related papers [23] and [24], dealing with boundary source terms.…”

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confidence: 99%

A potential well theory for the wave equation with nonlinear source and boundary damping terms

Vitillaro

2002

Glasgow Math. J.

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Abstract. The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is; 1Þ Â À 1 ; uð0; xÞ ¼ u 0 ðxÞ; u t ð0; xÞ ¼ u 1 ðxÞ on ;where & R n (n ! 1) is a regular and bounded domain,1 Þ, ! 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of , to this problem.

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“…See for example [2], [12], [13], [14], [15], [21], [22], [29], [34]. We also refer to the related papers [23] and [24], dealing with boundary source terms.…”

mentioning

confidence: 99%

A potential well theory for the wave equation with nonlinear source and boundary damping terms

Vitillaro

2002

Glasgow Math. J.

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“…The main conclusion in [5] states that if/ =fiu) and if there exists t/0 > 0 such that /, /' are both positive and increasing for u > T/n, then global solutions exist when i=r[Av)f'iv)yidn = oo (1.4) and the solution blows-up in finite time for a class of initial functions when 7 < oo. This nonexistence problem due to a positive nonlinear function on the boundary surface has also been discussed by Levine and Payne [2] and by Pao [3] using a different argument. In all of the above papers the nonexistence of global solution requires a sufficiently large initial function u0 in some sense.…”

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confidence: 63%

Asymptotic Behavior of Solutions for a Parabolic Equation with Nonlinear Boundary Conditions

Pao¹

1980

Proceedings of the American Mathematical Society

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Abstract. This paper is concerned with the asymptotic behavior and nonexistence of global solutions for a linear parabolic equation under nonlinear boundary conditions. It is shown under certain conditions on the nonlinear boundary function that a global solution exists and converges to a steady-state solution while for another class of initial function the solution blows-up in finite time. In some special cases, threshold results for the convergence of the solution and its blowingup behavior are explicitly given.

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“…All other results in 2.1 and 2.2 are proved by comparison with sub-and supersolutions of selfsimilar type. Blow-up of solutions emanating from "large" initial data was established in [LP1] using energy methods. In [Wa] both the global existence and the blow-up result were shown by comparison arguments.…”

Section: 10mentioning

confidence: 99%

Blow-up on the boundary: a survey

Фила

Filo

1996

Banach Center Publ.

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Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time

Cited by 214 publications

References 6 publications

A potential well theory for the wave equation with nonlinear source and boundary damping terms

A potential well theory for the wave equation with nonlinear source and boundary damping terms

Asymptotic Behavior of Solutions for a Parabolic Equation with Nonlinear Boundary Conditions

Blow-up on the boundary: a survey

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