Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities
Abstract
This research considers a class of quasi-linear parabolic equation with
variable exponents:%
\begin{equation*}
a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f\left( u\right)
\end{equation*}%
in which $a(x,t)>0$ is a nonnegative function and the exponents of
nonlinearity $m(x)$, $p(x)$ are given measurable functions. Under suitable
conditions on the given data a finite-time blow-up result of solutions is
proven if the initial datum possesses suitable positive energy and in this
case we precise estimate for the lifespan $T^{\ast }$ of the solution.
Blow-up of solutions with negative initial energy is also established.
Keywords
Full Text:
PDFReferences
E. Acerbi and G. Mingione, Regularity results for stationary eletrorheological fluids, Arch. Ration. Mech. Anal 164 (2002), 213259.
G. Akagi and M. Ôtani, Evolutions inclusions governed by subdifferentials in reflexive banach spaces, J. Evol. Equ. 4 (2004), 519-541.
S.N. Antonsev, Blow up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math 234 (2010), 2633-2645.
Aiguo Bao and Xianfa Song, Bounds for the blowup time of the solutions to quasi-linear parabolic problems, Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 65 (2014).
Quasilinear evolution equation involving the m(:)-Laplacian Operator
L. Diening, P. Hästo, P. Harjulehto, and M. Ruzicka, Lebesgue and sobolev spaces with variable exponents, in: Springer Lecture Notes, Springer-Verlag, Berlin, 2011 and 2017.
L. Diening and M. Ruzicka, Calderon Zygmund operators on generalized Lebesgue spaces Lp(x) and problems related to fluid dynamics, Preprint Mathematische
Fakultät, Albert-Ludwigs-Universität Freiburg.
X. Fan, J. Shen, and D. Zhao, Sobolev embedding theorems for spaces
Wk;p(x)(), J. Math. Anal. Appl. 262 (2001), 749-760.
Y. Fu, The existence of solutions for elliptic systems with nonuniform growth, Studia Math. 151 (2002), 227246.
H. Fujita, On the blowing up of solutions of the Cauchy problem for, J. Fac. Sci. Univ. Tokyo Sect. 13 (1966), no. I, 109-124.
V. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equation of paprabolic and hyperbolic types, J. Sov. Math. 10 (1978), 53-70.
Baghaei Khadijeh, Ghaemi Mohammad Bagher, and Hesaaraki Mahmoud,
Lower bounds for the blow-up time in a semilinear parabolic problem involving a variable source, Applied Mathematics Letters 27 (2014).
O. Kovàcik and J. Rákosnik, On spaces lp(x) and w1;p(x)(!), vol. 41, 1991.
W. M. Ni, P. E. Sacks, and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), 97-120.
L. E. Payne, Improperly posed problems in partial dierential equations. Regional Conference Series in AppliedMathematics, 1975
Ferreira R., de Pablo A., Pérez-LLanos M., and Rossi J. D., Critical exponents for a semilinear parabolic equation with variable reaction, Proceedings of the Royal Society of Edinburgh Section A Mathematics 142 (2012).
Zhong Tan, The reaction-diusion equation with lewis function and critical sobolev exponent, Journal of Mathematical Analysis and Applications 272 (2002), no. 2, 480-495.
Hua Wang and Yijun He, On blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Applied Mathematics Letters 26 (2013), no. 10, 1008 -1012.
Wu Xiulan, Guo Bin, and Gao Wenjie, Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Applied Mathematics Letters 26 (2013).
Yong Zhou, Global nonexistence for a quasilinear evolution equation with critical lower energy, Arch. Inequal. Appl. 2 (2004), 41-47.20. , Global nonexistence for a quasilinear evolution equation with a gen-
eralized lewis function, Journal for Analysis and its Applications 24 (2005), 179-187.
DOI: http://dx.doi.org/10.24193/subbmath.2021.3.11
Refbacks
- There are currently no refbacks.