Abstract
AbstractIn this paper, we consider the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations. Cauchy problems for higher order parabolic equations have no positivity preserving property in general, however, it is expected that solutions to these Cauchy problems are eventually globally positive if initial data decay slowly enough. We first show the existence of the threshold of the decay rate of initial datum which separates whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. Applying this result, we construct eventually globally positive solutions to the Cauchy problem of the semilinear polyharmonic heat equation under the super-Fujita condition.
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Berchio, E.: On the sign of solutions to some linear parabolic biharmonic equations. Adv. Differ. Equ. 13(9–10), 959–976 (2008)
2. Bernis, F.: Change of sign of the solutions to some parabolic problems. In: Nonlinear Analysis and Applications (Arlington, Tex., 1986). Lecture Notes in Pure and Appl. Math., vol. 109, pp. 75–82. Dekker, New York (1987)
3. Caristi, G., Mitidieri, E.: Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data. J. Math. Anal. Appl. 279(2), 710–722 (2003). https://doi.org/10.1016/S0022-247X(03)00062-3
4. Coffman, C.V., Grover, C.L.: Obtuse cones in Hilbert spaces and applications to partial differential equations. J. Funct. Anal. 35(3), 369–396 (1980). https://doi.org/10.1016/0022-1236(80)90088-9
5. Egorov, Y.V., Galaktionov, V.A., Kondratiev, V.A., Pohozaev, S.I.: On the necessary conditions of global existence to a quasilinear inequality in the half-space. C. R. Acad. Sci. Paris Sér. I Math. 330(2), 93–98 (2000). https://doi.org/10.1016/S0764-4442(00)00124-5