TY - JOUR

T1 - Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

AU - Grunau, Hans Christoph

AU - Miyake, Nobuhito

AU - Okabe, Shinya

N1 - Funding Information:
This work was initiated during the first author’s visit at Tohoku University. The first author is very grateful to second and third authors for their warm hospitality and the inspiring working atmosphere. The second author was supported in part by the Grant-in-Aid for JSPS Fellows (No. JP19J10424) from the Japan Society for the Promotion of Science. The third author was supported in part by the Grant-in-Aid for Scientific Research(S) (No. JP19H05599) from the Japan Society for the Promotion of Science.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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.

AB - 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.

KW - biharmonic heat equations

KW - global positivity

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U2 - 10.1515/anona-2020-0138

DO - 10.1515/anona-2020-0138

M3 - Article

AN - SCOPUS:85091583406

VL - 10

SP - 353

EP - 370

JO - Advances in Nonlinear Analysis

JF - Advances in Nonlinear Analysis

SN - 2191-9496

IS - 1

ER -