Finite extinction time for a nonlinear parabolic Neumann boundary value problem. (English) Zbl 0833.35076
Let \(u\) be a nonnegative, bounded solution of the nonlinear parabolic equation \(u_t= \Delta\varphi(u)- f(u)\) on \(\Omega\times (0, \infty)\) (\(\Omega\) bounded) with homogeneous Neumann boundary conditions and a nontrivial, nonnegative initial condition. We show that there exists a time \(t_0> 0\) for which \(u(x, t)= 0\) for all \(t\geq t_0\) if and only if \(\int^\varepsilon_0 {ds\over f(s)}< \infty\).
Reviewer: A.V.Lair (Wright-Patterson)
MSC:
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
finite extinction timeReferences:
| [1] | DOI: 10.1512/iumj.1981.30.30056 · Zbl 0598.76100 · doi:10.1512/iumj.1981.30.30056 |
| [2] | DOI: 10.1007/BF01448367 · Zbl 0652.35043 · doi:10.1007/BF01448367 |
| [3] | Crandall M. G., Proceedings of Symposia in Pure Mathematics 45 (1986) |
| [4] | DOI: 10.1080/03605307908820126 · Zbl 0425.35057 · doi:10.1080/03605307908820126 |
| [5] | Diaz I., Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid 72 pp 613– (1978) |
| [6] | DOI: 10.1016/0041-5553(74)90073-1 · Zbl 0306.35060 · doi:10.1016/0041-5553(74)90073-1 |
| [7] | DOI: 10.1137/0143085 · Zbl 0536.35039 · doi:10.1137/0143085 |
| [8] | Ladyzenskaja O. A., AMS Transl. Math. MoNo. 23 (1968) |
| [9] | DOI: 10.1016/0362-546X(93)90172-O · Zbl 0833.35075 · doi:10.1016/0362-546X(93)90172-O |
| [10] | DOI: 10.1137/0119026 · Zbl 0207.10301 · doi:10.1137/0119026 |
| [11] | Protter M. H., Maximum Principles in Differential Equations (1967) · Zbl 0153.13602 |
| [12] | DOI: 10.1016/0362-546X(88)90106-X · Zbl 0674.35050 · doi:10.1016/0362-546X(88)90106-X |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
