Critical exponents for the evolution p-Laplacian equation with a localized reaction

Zhilei Liang ()
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Zhilei Liang: Southwestern University of Finance and Economics

Indian Journal of Pure and Applied Mathematics, 2012, vol. 43, issue 5, 535-558

Abstract: Abstract This paper deals with the large time behavior of nonnegative solutions to the equation $$u_t = div\left( {\left| {\nabla u} \right|^{p - 2} \nabla u} \right) + a\left( x \right)u^q ,\left( {x,t} \right) \in R^N \times (0,T),$$ where p > 2, q > 0, and the function a(x) ≥ 0 has a compact support. We obtain the critical exponent for global existence q 0 and the Fujita exponent q c . In one-dimensional case N = 1, we have $$q_0 = \frac{{2(p - 1)}} {p}$$ and q c = 2(p − 1). Particularly, all solutions are global in time if 0 q c both blowing up solutions and global solutions exist. However, for the case N ≥ p > 2, these two critical exponents are exactly the same. Namely, q 0 = p − 1 = q c .

Keywords: p-Laplacian equation; global existence; blowing up; localized reaction (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s13226-012-0032-1

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