Flow invariance for perturbed nonlinear evolution equations
Let X be a real Banach space, J=[0,a]⊂R, A:D(A)⊂X→2X\ϕ an m-accretive operator and f:J×X→X continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets K⊂X for the evolution system u′+Au∍f(t,u) on J=[0,a]. More generall...
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| Format: | Article |
| Language: | English |
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Wiley
1996-01-01
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| Series: | Abstract and Applied Analysis |
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| Online Access: | http://dx.doi.org/10.1155/S1085337596000231 |
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| _version_ | 1832548779341905920 |
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| author | Dieter Bothe |
| author_facet | Dieter Bothe |
| author_sort | Dieter Bothe |
| collection | DOAJ |
| description | Let X be a real Banach space, J=[0,a]⊂R, A:D(A)⊂X→2X\ϕ an m-accretive operator and f:J×X→X continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets K⊂X for the evolution system
u′+Au∍f(t,u) on J=[0,a].
More generally, we provide conditions under which this evolution system has
mild solutions satisfying time-dependent constraints u(t)∈K(t) on J. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of type ut=ΔΦ(u)+g(u) in (0,∞)×Ω, Φ(u(t,⋅))|∂Ω=0, u(0,⋅)=u0 under certain assumptions on the setΩ⊂Rn the function Φ(u1,…,um)=(φ1(u1),…,φm(um)) and g:R+m→Rm. |
| format | Article |
| id | doaj-art-aef13a8b16304fd0967e8304e44fa851 |
| institution | Kabale University |
| issn | 1085-3375 |
| language | English |
| publishDate | 1996-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-aef13a8b16304fd0967e8304e44fa8512025-02-03T06:13:07ZengWileyAbstract and Applied Analysis1085-33751996-01-011441743310.1155/S1085337596000231Flow invariance for perturbed nonlinear evolution equationsDieter Bothe0Fachbereich 17, Universität Paderborn, Paderborn D-33095, GermanyLet X be a real Banach space, J=[0,a]⊂R, A:D(A)⊂X→2X\ϕ an m-accretive operator and f:J×X→X continuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed sets K⊂X for the evolution system u′+Au∍f(t,u) on J=[0,a]. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraints u(t)∈K(t) on J. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of type ut=ΔΦ(u)+g(u) in (0,∞)×Ω, Φ(u(t,⋅))|∂Ω=0, u(0,⋅)=u0 under certain assumptions on the setΩ⊂Rn the function Φ(u1,…,um)=(φ1(u1),…,φm(um)) and g:R+m→Rm.http://dx.doi.org/10.1155/S1085337596000231Nonlinear evolution equationtime-dependent constraintsviabilityreaction-diffusion system global existence. |
| spellingShingle | Dieter Bothe Flow invariance for perturbed nonlinear evolution equations Abstract and Applied Analysis Nonlinear evolution equation time-dependent constraints viability reaction-diffusion system global existence. |
| title | Flow invariance for perturbed nonlinear evolution equations |
| title_full | Flow invariance for perturbed nonlinear evolution equations |
| title_fullStr | Flow invariance for perturbed nonlinear evolution equations |
| title_full_unstemmed | Flow invariance for perturbed nonlinear evolution equations |
| title_short | Flow invariance for perturbed nonlinear evolution equations |
| title_sort | flow invariance for perturbed nonlinear evolution equations |
| topic | Nonlinear evolution equation time-dependent constraints viability reaction-diffusion system global existence. |
| url | http://dx.doi.org/10.1155/S1085337596000231 |
| work_keys_str_mv | AT dieterbothe flowinvarianceforperturbednonlinearevolutionequations |