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

Grunau, Hans-Christoph; Miyake, Nobuhito; Okabe, Shinya

doi:10.1515/anona-2020-0138

Cited by 8 publications

(5 citation statements)

References 14 publications

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“…by a formal approach based on spectral analysis. Similar consideration can been found in [12,21]. In this paper, we derive the L ∞ decay estimate like (1.7) for the solutions of problems (1.1) and (1.2).…”

Section: Introductionsupporting

confidence: 59%

$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

Wang

Chen

2021

Bound Value Probl

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In this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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Section: Introductionsupporting

confidence: 59%

$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

Wang

Chen

2021

Bound Value Probl

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“…On the other hand, we are con dent the tools introduced here may reveal useful also in di erent higher order contexts, such as parabolic problems, in the study of the sign of solutions to quasilinear equations and in the higher order fractional Laplacian setting [4,8,14]. This research started in 2010 when Theorem Notation.…”

Section: Overviewmentioning

confidence: 92%

Maximum principle for higher order operators in general domains

Cassani

Tarsia

2021

Advances in Nonlinear Analysis

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We first prove De Giorgi type level estimates for functions in W 1,t (Ω), Ω ⊂ R N $ \Omega\subset{\mathbb R}^N $ , with t > N ≥ 2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2(Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

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“…For the fractional Laplacian, that is for α ∈ (0, 1), the positivity is obtained directly from the fractional heat kernels, as shown in [36,Section 2]. In all other cases we can only expect (local) eventual positivity, unless we restrict to special classes of initial conditions as shown in [26].…”

Section: Local Eventual Positivity Of Solutionsmentioning

confidence: 99%

Local uniform convergence and eventual positivity of solutions to biharmonic heat equations

Daners¹,

Glück²,

Mui³

2021

Preprint

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We study the evolution equation associated with the biharmonic operator on infinite cylinders with bounded smooth cross-section subject to Dirichlet boundary conditions. The focus is on the asymptotic behaviour and positivity properties of the solutions for large times. In particular, we derive the local eventual positivity of solutions. We furthermore prove the local eventual positivity of solutions to the biharmonic heat equation and its generalisations on Euclidean space. The main tools in our analysis are the Fourier transform and spectral methods.

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

Cited by 8 publications

References 14 publications

$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

$L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

Maximum principle for higher order operators in general domains

Local uniform convergence and eventual positivity of solutions to biharmonic heat equations

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