Victor A. Galaktionov, "Critical global asymptotics in higher-order semilinear parabolic equations", International Journal of Mathematics and Mathematical Sciences, vol. 2003, Article ID 406190, 17 pages, 2003. https://doi.org/10.1155/S0161171203210176
Critical global asymptotics in higher-order semilinear parabolic equations
We consider a higher-order semilinear parabolic equation in , . The nonlinear term is homogeneous: and for any , with exponents , and . We also assume that 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 such that the asymptotic behavior as of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solution of the parabolic operator , so that for , , where is a constant depending on , , and only.
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