TY - JOUR
T1 - Life-Spans and Blow-Up Rates for a $p$-Laplacian Parabolic Equation with General Source
AU - Long , Qunfei
JO - Journal of Partial Differential Equations
VL - 2
SP - 187
EP - 197
PY - 2024
DA - 2024/06
SN - 37
DO - http://doi.org/10.4208/jpde.v37.n2.5
UR - https://global-sci.org/intro/article_detail/jpde/23208.html
KW - $p$-Laplacian, parabolic equation, blow-ups, life-spans, blow-up rates.
AB - <p style="text-align: justify;">This article investigates the blow-up results for the initial boundary value
problem to the quasi-linear parabolic equation with $p$-Laplacian $$u_t−∇·( |∇u|^{p−2}∇u)= f(u),$$ where $p≥2$ and the function $f(u)$ satisfies $$α\int^u_0f(s)ds≤u f(u)+βu^p+\gamma, u&gt;0$$ for some positive constants $α,β,\gamma$ with $0&lt;β≤ \frac{(α−p)λ_{1,p}}{p},$ which has been studied under
the initial condition $J_p(u_0)&lt;0.$ This paper generalizes the above results on the following aspects: a new blow-up condition is given, which holds for all $p&gt;2;$ a new blow-up
condition is given, which holds for $p=2;$ some new lifespans and upper blow-up rates
are given under certain conditions.</p>