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Volume 24, Issue 2
Blow-up of Solutions for a Singular Nonlocal Viscoelastic Equation

Shun Tang Wu

DOI: 10.4208/jpde.v24.n2.3

J. Part. Diff. Eq., 24 (2011), pp. 140-149.

Published online: 2011-05

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  • Abstract

We study the nonlinear one-dimensional viscoelastic nonlocal problem:  $u_{tt}-\frac{1}{x}(xu_x)_x+ ∫^t_0g(t-s)\frac{1}{x}(xu_x(x,s))_xds=|u|^{p-2}u$, with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].

  • Keywords

Blow-up life span viscoelastic nonlocal problem

  • AMS Subject Headings

35B35 35L70 35L20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-24-140, author = {Shun Tang Wu }, title = {Blow-up of Solutions for a Singular Nonlocal Viscoelastic Equation}, journal = {Journal of Partial Differential Equations}, year = {2011}, volume = {24}, number = {2}, pages = {140--149}, abstract = {

We study the nonlinear one-dimensional viscoelastic nonlocal problem:  $u_{tt}-\frac{1}{x}(xu_x)_x+ ∫^t_0g(t-s)\frac{1}{x}(xu_x(x,s))_xds=|u|^{p-2}u$, with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v24.n2.3}, url = {http://global-sci.org/intro/article_detail/jpde/5202.html} }
TY - JOUR T1 - Blow-up of Solutions for a Singular Nonlocal Viscoelastic Equation AU - Shun Tang Wu JO - Journal of Partial Differential Equations VL - 2 SP - 140 EP - 149 PY - 2011 DA - 2011/05 SN - 24 DO - http://doi.org/10.4208/jpde.v24.n2.3 UR - https://global-sci.org/intro/article_detail/jpde/5202.html KW - Blow-up KW - life span KW - viscoelastic KW - nonlocal problem AB -

We study the nonlinear one-dimensional viscoelastic nonlocal problem:  $u_{tt}-\frac{1}{x}(xu_x)_x+ ∫^t_0g(t-s)\frac{1}{x}(xu_x(x,s))_xds=|u|^{p-2}u$, with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].

Shun Tang Wu . (2011). Blow-up of Solutions for a Singular Nonlocal Viscoelastic Equation. Journal of Partial Differential Equations. 24 (2). 140-149. doi:10.4208/jpde.v24.n2.3
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