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Large time behavior of solutions for the porous medium equation with a nonlinear gradient source

Nan Li1, Pan Zheng23*, Chunlai Mu2 and Iftikhar Ahmed2

Author Affiliations

1 Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, P.R. China

2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China

3 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China

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Boundary Value Problems 2014, 2014:98  doi:10.1186/1687-2770-2014-98

Published: 6 May 2014

Abstract

This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>, we prove that the global solution converges to the separate variable solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M6">View MathML</a>. While <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M7">View MathML</a>, we note that the global solution converges to the separate variable solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8">View MathML</a>. Moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>, we show that the global solution also converges to the separate variable solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8">View MathML</a> for the small initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a>, and we find that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> blows up in finite time for the large initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a>.

MSC: 35K55, 35K65, 35B40.

Keywords:
large time behavior; separate variable solution; porous medium equation; gradient source; blow-up