Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone |
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Authors: | Changchun Liu |
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Institution: | 1.Department of Mathematics,Jilin University,Changchun,China |
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Abstract: | In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u t = Δu m + V(x)h(t)u p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2 + (n ? 2)l = ω 1. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u 0 ≥ 0. |
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