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Volume 37, Issue 3
Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Fang Li & Jingjing Zhang

DOI: 10.4208/jpde.v37.n3.3

J. Part. Diff. Eq., 37 (2024), pp. 263-277.

Published online: 2024-08

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  • Abstract
In this paper, we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms $$A(t)|u_{t}|^{m-2}u_{t}-\Delta u+\int_0^{t}g(t-s)\Delta u(x,s){\rm d}s=|u|^{p-2}u\log |u|.$$ Due to the presence of the log source term, it is not possible to use the source term to dominate the term $A(t)|u_{t}|^{m-2}u_{t}$. To bypass this difficulty, we build up inverse Hölder-like inequality and then apply differential inequality argument to prove the solution blows up in finite time. In addition, we can also give a decay rate under a general assumption on the relaxation functions satisfying $g'\leq -\xi(t)H(g(t)),~H(t)=t^\nu,~t\geq 0,~\nu>1$. This improves the existing results.
  • Keywords

Viscoelastic term, blow up, decay estimate.

  • AMS Subject Headings

35B44, 35K51, 35A01

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-263, author = {Li , Fang and Zhang , Jingjing}, title = {Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {3}, pages = {263--277}, abstract = {
In this paper, we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms $$A(t)|u_{t}|^{m-2}u_{t}-\Delta u+\int_0^{t}g(t-s)\Delta u(x,s){\rm d}s=|u|^{p-2}u\log |u|.$$ Due to the presence of the log source term, it is not possible to use the source term to dominate the term $A(t)|u_{t}|^{m-2}u_{t}$. To bypass this difficulty, we build up inverse Hölder-like inequality and then apply differential inequality argument to prove the solution blows up in finite time. In addition, we can also give a decay rate under a general assumption on the relaxation functions satisfying $g'\leq -\xi(t)H(g(t)),~H(t)=t^\nu,~t\geq 0,~\nu>1$. This improves the existing results.
}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n3.3}, url = {http://global-sci.org/intro/article_detail/jpde/23342.html} }
TY - JOUR T1 - Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem AU - Li , Fang AU - Zhang , Jingjing JO - Journal of Partial Differential Equations VL - 3 SP - 263 EP - 277 PY - 2024 DA - 2024/08 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n3.3 UR - https://global-sci.org/intro/article_detail/jpde/23342.html KW - Viscoelastic term, blow up, decay estimate. AB -
In this paper, we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms $$A(t)|u_{t}|^{m-2}u_{t}-\Delta u+\int_0^{t}g(t-s)\Delta u(x,s){\rm d}s=|u|^{p-2}u\log |u|.$$ Due to the presence of the log source term, it is not possible to use the source term to dominate the term $A(t)|u_{t}|^{m-2}u_{t}$. To bypass this difficulty, we build up inverse Hölder-like inequality and then apply differential inequality argument to prove the solution blows up in finite time. In addition, we can also give a decay rate under a general assumption on the relaxation functions satisfying $g'\leq -\xi(t)H(g(t)),~H(t)=t^\nu,~t\geq 0,~\nu>1$. This improves the existing results.
Li , Fang and Zhang , Jingjing. (2024). Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem. Journal of Partial Differential Equations. 37 (3). 263-277. doi:10.4208/jpde.v37.n3.3
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