A Gaussian upper bound of the conjugate heat equation along Ricci-harmonic flow

We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “[Formula: see text]”, where [Formula: see text] is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.

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Gradient estimate for the forward conjugate heat equation of forms under the Ricci flow

International Journal of Mathematics ◽

10.1142/s0129167x21500816 ◽

2021 ◽

pp. 2150081

Author(s):

Liangdi Zhang

Keyword(s):

Riemannian Manifold ◽

Heat Equation ◽

Ricci Flow ◽

Differential Forms ◽

Gradient Estimate ◽

Conjugate Heat Equation

We establish bounds for the gradient of solutions to the forward conjugate heat equation of differential forms on a Riemannian manifold with the metric evolves under the Ricci flow.

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A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2022001 ◽

2022 ◽

Author(s):

Lijuan Wang ◽

Can Zhang

Keyword(s):

Heat Equation ◽

Upper Bound ◽

Unbounded Domains ◽

Null Controllability ◽

Nonlinear Pdes ◽

Heat Equations ◽

Semilinear Heat Equation ◽

Lipschitz Nonlinearity ◽

Exact Null Controllability ◽

Suitable Approximation

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R N . As an application, we then show the exact null-controllability for this semilinear heat equation in R N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in R N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.

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