Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

Abstract

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.

Blow-up of weak solutions for a higher-order heat equation with logarithmic nonlinearity

“…Recently many other authors investigated higher-order hyperbolic and parabolic type equation [5, 6, 8, 10, 13-15, 17, 18]. Ishige et al [8] studied the Cauchy problem for nonlinear higher-order heat equation as follows…”

Solvability of superlinear fractional parabolic equations

“…Results of this type are important for the development of any general function-analytic framework for well-posedness of problem (SHE), for example in identifying an optimal Orlicz class of initial data for a given nonlinearity f . Such studies exist for a variety of semilinear problems: see [6,8] for problem (SHE) with power law and power-log nonlinearities; see [10,11] for semilinear parabolic systems; [15] for the linear heat equation with nonlinear boundary conditions; [17,18] for the Hardy parabolic equation; [22] for higher-order semilinear parabolic equations; [16] for semilinear heat equations in a half-space of R N .…”

Blow Up at Infinity of Weak Solutions for a Higher-Order Parabolic Equation With Logarithmic Nonlinearity

“…Also, He et al [6] proved the decay and the finite time blow-up for weak solutions of the equation. Resently many other authors investigated higher-order hyperbolic and parabolic type equation [4,7,11,12,13,14,15]. Ishige et al [7] studied the Cauchy problem for nonlinear higher-order heat equation as follows…”