Positive solutions of a nonlinear boundary-value problem of parabolic type

Pao, C. V.

doi:10.1016/0022-0396(76)90008-5

Cited by 46 publications

(16 citation statements)

References 23 publications

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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: 62%

Asymptotic behavior of solutions for a parabolic equation with nonlinear boundary conditions

Pao¹

1980

Proc. Amer. Math. Soc.

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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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confidence: 62%

Asymptotic behavior of solutions for a parabolic equation with nonlinear boundary conditions

Pao¹

1980

Proc. Amer. Math. Soc.

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“…Sattinger [15] extended Amann's results (assuming that f(x, t, u) is C ' with respect to u) to parabolic initial boundary value problems using a similar monotone iteration scheme. This work was subsequently extended to various kinds of problems for both elliptic and parabolic equations by Pao [10], [11] and Puel [13] using either monotone iteration techniques or the theory of monotone operators.…”

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confidence: 99%

On the Existence of Maximal and Minimal Solutions for Parabolic Partial Differential Equations

Bebernes¹,

Schmitt²

1979

Proceedings of the American Mathematical Society

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Abstract. The existence of maximal and minimal solutions for initialboundary value problems and the Cauchy initial value problem associated with Lu -f(x, t, u, Vu) where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows / to be highly nonlinear, i.e., / locally Holder continuous with almost quadratic growth in |V«|.

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“…(See [11], [14] for a detailed discussion. See also [10] for the uniqueness problem.) Hence the asymptotic behavior of the solution can be determined through the construction of suitable upper and lower solutions.…”

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confidence: 99%

Nonexistence of Global Solutions and Bifurcation Analysis for a Boundary-Value Problem of Parabolic Type

Pao¹

1977

Proceedings of the American Mathematical Society

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Abstract. The aim of this paper is to present a bifurcation analysis on the existence of the nonexistence of a global solution for a semilinear parabolic equation and to characterize the local stability and the instability of the corresponding steady-state solutions. The bifurcation result can be described either by a parameter X for a fixed spatial domain Ü or by varying Í2 for a fixed A. The stability analysis gives a result which can be used to determine the stability or instability problem when the system possesses nonintersecting multiple steady-state solutions.

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Positive solutions of a nonlinear boundary-value problem of parabolic type

Cited by 46 publications

References 23 publications

Asymptotic behavior of solutions for a parabolic equation with nonlinear boundary conditions

Asymptotic behavior of solutions for a parabolic equation with nonlinear boundary conditions

On the Existence of Maximal and Minimal Solutions for Parabolic Partial Differential Equations

Nonexistence of Global Solutions and Bifurcation Analysis for a Boundary-Value Problem of Parabolic Type

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