Abstract
Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.
Highlights
This paper is concerned with the positivity of solutions to Cauchy problems for fourth order parabolic equations.We say that a parabolic Cauchy problem has a positivity preserving property if non-negative and nontrivial initial data always yield solutions which are positive in the whole space and for any positive time
This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part
The rst goal of this paper is to nd su cient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space
Summary
This paper is concerned with the positivity of solutions to Cauchy problems for fourth order parabolic equations.We say that a parabolic Cauchy problem has a positivity preserving property if non-negative and nontrivial initial data always yield solutions which are positive in the whole space and for any positive time. Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. The rst goal of this paper is to nd su cient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space.
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