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Volume 33, Issue 3
Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory

Jiahui Huang, Junli Yuan & Yan Zhao

DOI: 10.4208/jpde.v33.n3.5

J. Part. Diff. Eq., 33 (2020), pp. 249-260.

Published online: 2020-06

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  • Abstract

In this paper, we investigate a reaction-diffusion equation $u_t-du_{xx}=au+\int_{0}^{t}u^p(x,\tau){\rm d}\tau+k(x)$ with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if $\int_{-h_0}^{h_0}k(x)\psi_1 {\rm d}x$ is large enough, then the blowup occurs. Meanwhile we also prove when $T^*<+\infty$, the solution must blow up in finite time. On the other hand, we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently.

  • Keywords

Nonlinear memory, free boundary, blowup, asymptotic behavior.

  • AMS Subject Headings

35K20, 35R35, 92B05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

1353988290@qq.com (Jiahui Huang)

yjltg17@163.com (Junli Yuan)

2372075036@qq.com (Yan Zhao)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-33-249, author = {Huang , JiahuiYuan , Junli and Zhao , Yan}, title = {Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {33}, number = {3}, pages = {249--260}, abstract = {

In this paper, we investigate a reaction-diffusion equation $u_t-du_{xx}=au+\int_{0}^{t}u^p(x,\tau){\rm d}\tau+k(x)$ with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if $\int_{-h_0}^{h_0}k(x)\psi_1 {\rm d}x$ is large enough, then the blowup occurs. Meanwhile we also prove when $T^*<+\infty$, the solution must blow up in finite time. On the other hand, we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v33.n3.5}, url = {http://global-sci.org/intro/article_detail/jpde/17073.html} }
TY - JOUR T1 - Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory AU - Huang , Jiahui AU - Yuan , Junli AU - Zhao , Yan JO - Journal of Partial Differential Equations VL - 3 SP - 249 EP - 260 PY - 2020 DA - 2020/06 SN - 33 DO - http://doi.org/10.4208/jpde.v33.n3.5 UR - https://global-sci.org/intro/article_detail/jpde/17073.html KW - Nonlinear memory, free boundary, blowup, asymptotic behavior. AB -

In this paper, we investigate a reaction-diffusion equation $u_t-du_{xx}=au+\int_{0}^{t}u^p(x,\tau){\rm d}\tau+k(x)$ with double free boundaries. We study blowup phenomena in finite time and asymptotic behavior of time-global solutions. Our results show if $\int_{-h_0}^{h_0}k(x)\psi_1 {\rm d}x$ is large enough, then the blowup occurs. Meanwhile we also prove when $T^*<+\infty$, the solution must blow up in finite time. On the other hand, we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently.

Huang , JiahuiYuan , Junli and Zhao , Yan. (2020). Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory. Journal of Partial Differential Equations. 33 (3). 249-260. doi:10.4208/jpde.v33.n3.5
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