Critical global asymptotics in higher-order semilinear parabolic equations
We consider a higher-order semilinear parabolic equation ut=−(−Δ)mu−g(x,u) in ℝN×ℝ+, m>1. The nonlinear term is homogeneous: g(x,su)≡|s|p−1sg(x,u) and g(sx,u)≡|s|Qg(x,u) for any s∈ℝ, with exponents P>1, and Q>−2m. We also assume that g satisfies necessary coercivity and monotonicity conditi...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203210176 |
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| Summary: | We consider a higher-order semilinear parabolic equation ut=−(−Δ)mu−g(x,u) in ℝN×ℝ+, m>1. The nonlinear term is homogeneous: g(x,su)≡|s|p−1sg(x,u) and g(sx,u)≡|s|Qg(x,u)
for any s∈ℝ, with exponents P>1, and Q>−2m. We also
assume that g satisfies necessary coercivity and monotonicity
conditions for global existence of solutions with sufficiently
small initial data. The equation is invariant under a group of
scaling transformations. We show that there exists a critical
exponent P=1+(2m+Q)/N such that the asymptotic behavior as t→∞ of a class of global small solutions is not
group-invariant and is given by a logarithmic perturbation of the
fundamental solution b(x,t)=t−N/2mf(xt−1/2m) of the
parabolic operator ∂/∂t+(−Δ)m, so that for t≫1, u(x,t)=C0(ln t)−N/(2m+Q)[b(x,t)+o(1)], where C0 is a
constant depending on m, N, and Q only. |
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| ISSN: | 0161-1712 1687-0425 |