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Volume 52, Issue 2
Global Existence and Blow-Up in a $p(x)$-Laplace Equation with Dirichlet Boundary Conditions

Yuhua Jian & Zuodong Yang

DOI: 10.4208/jms.v52n2.19.01

J. Math. Study, 52 (2019), pp. 111-126.

Published online: 2019-05

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  • Abstract

This paper is devoted to a $p(x)$-Laplace equation with Dirichlet boundary. We obtain the existence of global solution to the problem by employing the method of potential wells. On the other hand, we show that the solution will blow up in finite time with $u_0 \not\equiv 0$ and nonpositive initial energy functional $J(u_0).$ By defining a positive function $F(t)$ and using the method of concavity we find an upper bound for the blow-up time.

  • Keywords

$p(x)$-Laplace equation, global weak solution, finite time blow-up, upper bounds.

  • AMS Subject Headings

35J25, 35J65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zdyang_jin@263.net (Zuodong Yang)

  • BibTex
  • RIS
  • TXT
@Article{JMS-52-111, author = {Jian , Yuhua and Yang , Zuodong}, title = {Global Existence and Blow-Up in a $p(x)$-Laplace Equation with Dirichlet Boundary Conditions}, journal = {Journal of Mathematical Study}, year = {2019}, volume = {52}, number = {2}, pages = {111--126}, abstract = {

This paper is devoted to a $p(x)$-Laplace equation with Dirichlet boundary. We obtain the existence of global solution to the problem by employing the method of potential wells. On the other hand, we show that the solution will blow up in finite time with $u_0 \not\equiv 0$ and nonpositive initial energy functional $J(u_0).$ By defining a positive function $F(t)$ and using the method of concavity we find an upper bound for the blow-up time.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v52n2.19.01}, url = {http://global-sci.org/intro/article_detail/jms/13154.html} }
TY - JOUR T1 - Global Existence and Blow-Up in a $p(x)$-Laplace Equation with Dirichlet Boundary Conditions AU - Jian , Yuhua AU - Yang , Zuodong JO - Journal of Mathematical Study VL - 2 SP - 111 EP - 126 PY - 2019 DA - 2019/05 SN - 52 DO - http://doi.org/10.4208/jms.v52n2.19.01 UR - https://global-sci.org/intro/article_detail/jms/13154.html KW - $p(x)$-Laplace equation, global weak solution, finite time blow-up, upper bounds. AB -

This paper is devoted to a $p(x)$-Laplace equation with Dirichlet boundary. We obtain the existence of global solution to the problem by employing the method of potential wells. On the other hand, we show that the solution will blow up in finite time with $u_0 \not\equiv 0$ and nonpositive initial energy functional $J(u_0).$ By defining a positive function $F(t)$ and using the method of concavity we find an upper bound for the blow-up time.

Jian , Yuhua and Yang , Zuodong. (2019). Global Existence and Blow-Up in a $p(x)$-Laplace Equation with Dirichlet Boundary Conditions. Journal of Mathematical Study. 52 (2). 111-126. doi:10.4208/jms.v52n2.19.01
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