scholarly journals Existence of Solutions for the Evolution -Laplacian Equation Not in Divergence Form

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Changchun Liu ◽  
Junchao Gao ◽  
Songzhe Lian
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximate Solutions ◽  
Divergence Form ◽  
Existence Of Weak Solutions ◽  
Uniform Estimates ◽  
Laplacian Equation

The existence of weak solutions is studied to the initial Dirichlet problem of the equation , with inf . We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

scholarly journals QUALITATIVE PROPERTIES FOR A SIXTH–ORDER THIN FILM EQUATION

2010 ◽  
Vol 15 (4) ◽  
pp. 457-471 ◽  
Author(s):  
Changchun Liu
Keyword(s):  
Thin Film ◽  
Weak Solutions ◽  
Approximate Solutions ◽  
Sixth Order ◽  
Classical Solutions ◽  
Qualitative Properties ◽  
Existence Of Weak Solutions ◽  
Uniform Estimates ◽  
Thin Film Equation ◽  
Silicon Based

In this article, the author studies the qualitative properties of weak solutions for a sixth‐order thin film equation, which arises in the industrial application of the isolation oxidation of silicon. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the expansion of the support of solutions are also discussed.

scholarly journals Quasilinear Dirichlet problems with competing operators and convection

2020 ◽  
Vol 18 (1) ◽  
pp. 1510-1517
Author(s):  
Dumitru Motreanu
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximation Procedure ◽  
Convection Term ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Variational Structure

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn 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.

Radiative effects for some bidimensional thermoelectric problems

2016 ◽  
Vol 5 (4) ◽  
Author(s):  
Luisa Consiglieri
Keyword(s):  
Steady State ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Divergence Form ◽  
Radiative Effects ◽  
Existence Of Weak Solutions ◽  
Nonlinear Radiation ◽  
Radiation Type

AbstractThere are two main objectives in this paper. One is to find sufficient conditions to ensure the existence of weak solutions for some bidimensional thermoelectric problems. At the steady-state, these problems consist of a coupled system of elliptic equations of the divergence form, commonly accomplished with nonlinear radiation-type conditions on at least a nonempty part of the boundary of a

scholarly journals Existence results for some nonlinear degenerate problems in the anisotropic spaces

2021 ◽  
Vol 39 (6) ◽  
pp. 53-66
Author(s):  
Mohamed Boukhrij ◽  
Benali Aharrouch ◽  
Jaouad Bennouna ◽  
Ahmed Aberqi
Keyword(s):  
Elliptic Problem ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Sign Condition ◽  
Degenerate Elliptic ◽  
Anisotropic Spaces ◽  
Nonlinear Degenerate

Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,where for $i=1,...,N$ $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and $H_i(x,u,\nabla u)$ is a nonlinear term without a sign condition. Under suitable conditions on $a_i$ and $H_i$, we prove the existence of weak solutions.

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.

QUASISTATIC EVOLUTION IN THE THEORY OF PERFECTLY ELASTO-PLASTIC PLATES PART I: EXISTENCE OF A WEAK SOLUTION

2009 ◽  
Vol 19 (02) ◽  
pp. 229-256 ◽  
Author(s):  
ALEXEY DEMYANOV
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Flow Rule ◽  
Variational Theory ◽  
Absolutely Continuous ◽  
Quasistatic Evolution ◽  
Existence Of Weak Solutions ◽  
Strong Formulation

The existence of weak solutions to the quasistatic problems in the theory of perfectly elasto-plastic plates is studied in the framework of the variational theory for rate-independent processes. Approximate solutions are constructed by means of incremental variational problems in spaces of functions with bounded hessian. The constructed weak solution is shown to be absolutely continuous in time. A strong formulation of the flow rule is obtained.

scholarly journals Existence of Solutions for NonhomogeneousA-Harmonic Equations with Variable Growth

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Yongqiang Fu ◽  
Lifeng Guo
Keyword(s):  
Weak Solutions ◽  
Lipschitz Domain ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Existence Of Weak Solutions

We study the following nonhomogeneousA-harmonic equations:d*A(x,du(x))+B(x,u(x))=0,    x∈Ω,  u(x)=0,  x∈∂Ω, whereΩ⊂ℝnis a bounded and convex Lipschitz domain,A(x,du(x))andB(x,u(x))satisfy somep(x)-growth conditions, respectively. We obtain the existence of weak solutions for the above equations in subspace𝔎01,p(x)(Ω,Λl-1)ofW01,p(x)(Ω,Λl-1).

EXISTENCE OF SOLUTIONS WITH MOVING PHASE BOUNDARIES IN THERMOELASTICITY

2008 ◽  
Vol 05 (03) ◽  
pp. 589-611 ◽  
Author(s):  
HARUMI HATTORI
Keyword(s):  
Phase Transitions ◽  
Euler Equations ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Entropy Condition ◽  
Phase Boundaries ◽  
Kinetic Relation ◽  
Two Phase ◽  
Existence Of Weak Solutions ◽  
Dynamic Phase

We discuss the existence of weak solutions with moving phase boundaries in thermoelasticity related to dynamic phase transitions. One of the goals is to study the dynamical consequence of the stable and metastable states defined in this paper. We use the entropy condition and the kinetic relation as the main admissibility criteria to study the above goals for the Euler equations with nonmonotone constitutive relation. We discuss the case where there are two noninteracting phase boundaries moving in the opposite directions. A modification to treat the case where the two phase boundaries collide is also discussed.

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