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 - <p style="text-align: justify;">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.</p>