scholarly journals Liouville-type theorems for certain degenerate and singular parabolic equations

2010 ◽  
Vol 348 (15-16) ◽  
pp. 873-877 ◽  
Author(s):  
Emmanuele DiBenedetto ◽  
Ugo Gianazza ◽  
Vincenzo Vespri
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenxiong Chen ◽  
Leyun Wu

Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.


2021 ◽  
Vol 6 (10) ◽  
pp. 10506-10522
Author(s):  
Fanqi Zeng ◽  

<abstract><p>In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $\end{document} </tex-math></disp-formula></p> <p>on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.</p></abstract>


Author(s):  
Ugo Gianazza ◽  
Naian Liao

Abstract We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of $p$-Laplacian type, with $p$ in the sub-critical range $\big(1,\frac{2N}{N+1}\big]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.


2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Khalid Atifi ◽  
Idriss Boutaayamou ◽  
Hamed Ould Sidi ◽  
Jawad Salhi

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.


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