Abstract
Abstract We consider the higher-order semilinear parabolic equation $$\begin{array}{} \displaystyle \partial_t u = -(-{\it\Delta})^{m} u + u|u|^{p-1}, \end{array}$$ in the whole space ℝN, where p > 1 and m ≥ 1 is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m = 1, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [15].
Highlights
We are interested in the semilinear parabolic equation∂tu = Amu + u|u|p−, u( ) = u Am ≡ −(−∆)m, (1.1)where u(t) : RN → R with N ≥, ∆ stands for the standard Laplace operator in RN, and the exponents p and m are xed, p > and m ∈ N, m ≥ odd.The higher-order semilinear parabolic equation (1.1) is a natural generation of the classical semilinear heat equation (m = )
In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m =, which consists in two steps: the reduction of the problem to a nite dimensional one, solving the nite dimensional problem by a classical topological argument based on the index theory
Where u(t) : RN → R with N ≥, ∆ stands for the standard Laplace operator in RN, and the exponents p and m are xed, p > and m ∈ N, m ≥ odd
Summary
Where u(t) : RN → R with N ≥ , ∆ stands for the standard Laplace operator in RN, and the exponents p and m are xed, p > and m ∈ N, m ≥ odd. The higher-order semilinear parabolic equation (1.1) is a natural generation of the classical semilinear heat equation (m = ). It arises in many physical applications such as theory of thin lm, lubrication, convectionexplosion, phase translation, or applications to structural mechanics (see the Petetier-Troy book [28] and references therein). By standard results the local Cauchy problem for equation (1.1) can be solved in L ∩ L∞ thanks to the integral representation t u(t) = Km(t) * u + Km(t − s) * u(s)|u(s)|p− ds,
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