Global Sign-changing Solutions of a Higher Order Semilinear Heat Equation in the Subcritical Fujita Range

Abstract

Abstract A detailed study of two classes of oscillatory global (and blow-up) solutions was began in [20] for the semilinear heat equation in the subcritical Fujita range with bounded integrable initial data u(x, 0) = u0(x). This study is continued and extended here for the 2mth-order heat equation, for m ≥ 2, with non-monotone nonlinearity with the same initial data u0. The fourth order biharmonic case m = 2 is studied in greater detail. The blow-up Fujita-type result for (0.2) now reads as follows: blow-up occurs for any initial data u0 with positive first Fourier coefficient: ∫ u0(x) dx > 0, i.e., as for (0.1), any such arbitrarily small initial function u0(x) leads to blow-up. The construction of two countable families of global sign changing solutions is performed on the basis of bifurcation/branching analysis and a further analytic-numerical study. In particular, a countable sequence of bifurcation points of similarity solutions is obtained: