Finite time blow-up for a class of parabolic or pseudo-parabolic equations
In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations: ut – aΔut − Δu+bu = k(t) |u|p−2u, (x,t) ∈ Ω× (0,T), where a ≥ 0, b >−ł1 with ł1 being the principal eigenvalue for −Δ on H01 (Ω) and k(t) > 0. By using the potential well met...
| Main Authors: | , , |
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| Format: | Journal Article |
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Pergamon Press
2018
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| Online Access: | http://hdl.handle.net/20.500.11937/67734 |
| _version_ | 1848761644092489728 |
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| author | Sun, F. Liu, Lishan Wu, Yong Hong |
| author_facet | Sun, F. Liu, Lishan Wu, Yong Hong |
| author_sort | Sun, F. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:
ut – aΔut − Δu+bu = k(t) |u|p−2u, (x,t) ∈ Ω× (0,T),
where a ≥ 0, b >−ł1 with ł1 being the principal eigenvalue for −Δ on H01 (Ω) and k(t) > 0. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) J (u0;0) < 0; (ii) J (u0; 0) ≤ d (∞), where d (∞)is a nonnegative constant; (iii) 0 < J (u0;0) ≤ Cρ (0), where ρ (0) involves the L2-norm or H01-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level. |
| first_indexed | 2025-11-14T10:34:57Z |
| format | Journal Article |
| id | curtin-20.500.11937-67734 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T10:34:57Z |
| publishDate | 2018 |
| publisher | Pergamon Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-677342018-07-11T06:12:26Z Finite time blow-up for a class of parabolic or pseudo-parabolic equations Sun, F. Liu, Lishan Wu, Yong Hong In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations: ut – aΔut − Δu+bu = k(t) |u|p−2u, (x,t) ∈ Ω× (0,T), where a ≥ 0, b >−ł1 with ł1 being the principal eigenvalue for −Δ on H01 (Ω) and k(t) > 0. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) J (u0;0) < 0; (ii) J (u0; 0) ≤ d (∞), where d (∞)is a nonnegative constant; (iii) 0 < J (u0;0) ≤ Cρ (0), where ρ (0) involves the L2-norm or H01-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level. 2018 Journal Article http://hdl.handle.net/20.500.11937/67734 10.1016/j.camwa.2018.02.025 Pergamon Press restricted |
| spellingShingle | Sun, F. Liu, Lishan Wu, Yong Hong Finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| title | Finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| title_full | Finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| title_fullStr | Finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| title_full_unstemmed | Finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| title_short | Finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| title_sort | finite time blow-up for a class of parabolic or pseudo-parabolic equations |
| url | http://hdl.handle.net/20.500.11937/67734 |