scholarly journals Extinction behavior for a parabolic $ p $-Laplacian equation with gradient source and singular potential

2021 ◽  
Vol 7 (1) ◽  
pp. 915-924
Author(s):  
Dengming Liu ◽  
◽  
Luo Yang
Keyword(s):  
Weak Solution ◽  
Sobolev Inequality ◽  
Singular Potential ◽  
Energy Estimate ◽  
Differential Inequalities ◽  
Laplacian Equation ◽  
Solution Methods

<abstract><p>We concern with the extinction behavior of the solution for a parabolic $ p $-Laplacian equation with gradient source and singular potential. By energy estimate approach, Hardy-Littlewood-Sobolev inequality, a series of ordinary differential inequalities, and super-solution and sub-solution methods, we obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.</p></abstract>

scholarly journals Global Existence and Extinction Singularity for a Fast Diffusive Polytropic Filtration Equation with Variable Coefficient

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Dengming Liu ◽  
Changyu Liu
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Sobolev Inequality ◽  
Variable Coefficient ◽  
Existence Result ◽  
Energy Estimate ◽  
Differential Inequalities ◽  
Filtration Equation ◽  
Global Existence Result

In this article, we deal with an inhomogeneous fast diffusive polytropic filtration equation. By using the energy estimate approach, Hardy–Littlewood–Sobolev inequality, and a series of ordinary differential inequalities, we prove the global existence result and obtain the conditions on the occurrence of the extinction phenomenon of the weak solution.

Applications of monotone operators to a class of fractional Laplacian equation

2015 ◽  
Vol 26 (07) ◽  
pp. 1550043
Author(s):  
V. Raghavendra ◽  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Differential Operator ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Monotone Operators ◽  
Bounded Subset ◽  
Laplacian Equation ◽  
Continuous Boundary ◽  
Open Bounded Subset ◽  
Fractional Type

In this study we establish the existence of a weak solution for a class of nonlocal problem [Formula: see text] where [Formula: see text] is a general nonlocal integro-differential operator of fractional type, λ is a real parameter, Ω is an open bounded subset of ℝn(n > 2s, where s ∈(0, 1) is fixed) with continuous boundary ∂Ω. Here f, g1: Ω → ℝ and h : ℝ → ℝ are functions satisfying suitable hypotheses.

scholarly journals Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Huxiao Luo ◽  
Shengjun Li ◽  
Xianhua Tang
Keyword(s):  
Weak Solution ◽  
Nontrivial Solution ◽  
Perturbation Methods ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Weak Limit ◽  
Main Difficulty ◽  
Potential Term ◽  
Laplacian Equation ◽  
Laplacian Equations

We study the existence of nontrivial solution of the following equation without compactness: (-Δ)pαu+up-2u=f(x,u),  x∈RN, where N,p≥2,  α∈(0,1),  (-Δ)pα is the fractional p-Laplacian, and the subcritical p-superlinear term f∈C(RN×R) is 1-periodic in xi for i=1,2,…,N. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional p-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution uλ of perturbation equations. And we prove that uλ→u as λ→0. Finally, by using vanishing lemma and periodic condition, we get that u is a nontrivial solution of fractional p-Laplacian equation.

scholarly journals EXTINCTION AND NON-EXTINCTION OF SOLUTIONS TO A P-LAPLACE EQUATION WITH A NONLOCAL SOURCE AND AN ABSORPTION TERM

2014 ◽  
Vol 19 (2) ◽  
pp. 169-179 ◽  
Author(s):  
Yuzhu Han ◽  
Haixia Li ◽  
Wenjie Gao
Keyword(s):  
Laplace Equation ◽  
Finite Time ◽  
Energy Estimate ◽  
Short Paper ◽  
Nonlocal Source ◽  
Solution Methods

In this short paper, the authors investigate the extinction and non-extinction of solutions to a fast diffusive p-Laplace equation with a nonlocal source and an absorption term. By applying the super-solution and sub-solution methods, instead of energy estimate methods, they give a classification of the exponents and coefficients for the solutions to vanish in finite time or not, which generalizes some previous results.

scholarly journals On a weak solution matching up with the double degenerate parabolic equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Initial Value ◽  
Laplacian Equation ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Regularized Method ◽  
Increasing Function

Abstract The well-posedness of weak solutions to a double degenerate evolutionary $p(x)$ p ( x ) -Laplacian equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)\bigr), $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) ) , is studied. It is assumed that $b(x,t)| _{(x,t)\in \varOmega \times [0,T]}>0$ b ( x , t ) | ( x , t ) ∈ Ω × [ 0 , T ] > 0 but $b(x,t) | _{(x,t)\in \partial \varOmega \times [0,T]}=0$ b ( x , t ) | ( x , t ) ∈ ∂ Ω × [ 0 , T ] = 0 , $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , and $A(s)$ A ( s ) is a strictly monotone increasing function with $A(0)=0$ A ( 0 ) = 0 . A weak solution matching up with the double degenerate parabolic equation is introduced. The existence of weak solution is proved by a parabolically regularized method. The stability theorem of weak solutions is established independent of the boundary value condition. In particular, the initial value condition is satisfied in a wider generality.

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

2019 ◽  
Vol 19 (1) ◽  
pp. 197-217 ◽  
Author(s):  
Caifeng Zhang
Keyword(s):  
Weak Solution ◽  
Lower Bound ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Existence Of Weak Solutions ◽  
Laplacian Equation ◽  
Trudinger Inequality ◽  
Dimension One

Abstract In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in {W^{s,p}(\mathbb{R}^{N})} . Define \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau>0. There holds \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}<+\infty, where {s\in(0,1)} , {sp=N} , {\alpha\in[0,\alpha_{*})} and \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where {V(x)} has a positive lower bound, {f(x,t)} behaves like {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and {\varepsilon>0} . Moreover, we also derive a weak solution with negative energy.

scholarly journals On the absence of global solutions to two-times-fractional differential inequalities involving Hadamard-Caputo and Caputo fractional derivatives

2022 ◽  
Vol 7 (4) ◽  
pp. 5830-5843
Author(s):  
Ibtehal Alazman ◽  
◽  
Mohamed Jleli ◽  
Bessem Samet ◽  
Keyword(s):  
Global Solution ◽  
Differential Inequality ◽  
Fractional Derivatives ◽  
Singular Potential ◽  
Global Solutions ◽  
Differential Inequalities ◽  
Caputo Fractional Derivatives ◽  
Potential Term ◽  
Fractional Differential ◽  
Derivatives Of

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>

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