Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model

Zijuan Wen; Meng Fan; Asim M. Asiri; Ebraheem O. Alzahrani; Mohamed M. El-Dessoky; Yang Kuang

doi:10.3934/mbe.2017025

Article Contents

2017, Volume 14, Issue 2: 407-420. Doi: 10.3934/mbe.2017025

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Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2.
School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, China
3.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
4.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA

¹Meng Fan is partially supported by NSFC-11271065, RFPD-20130043110001, and RFCPCMSP-2014

Received: January 26, 2016

Revised: June 27, 2016

Published: July 2016

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Abstract

This paper studies the global existence and uniqueness of classicalsolutions for a generalized quasilinear parabolic equation withappropriate initial and mixed boundary conditions. Under somepracticable regularity criteria on diffusion item and nonlinearity, weestablish the local existence and uniqueness of classical solutionsbased on a contraction mapping. This local solution can be continuedfor all positive time by employing the methods of energy estimates, $ L^{p} $-theory, and Schauder estimate of linear parabolic equations. Astraightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitroglioblastoma growth is also presented.

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Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09; Second: 92B05.

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