scholarly journals Well-posedness for nonlinear SPDEs with strongly continuous perturbation

2020 ◽  
pp. 1-31
Author(s):  
Guy Vallet ◽  
Aleksandra Zimmermann
Keyword(s):  
Stochastic Perturbation ◽  
Dirichlet Boundary Conditions ◽  
Well Posedness ◽  
Lipschitz Continuous ◽  
Bounded Lipschitz Domain ◽  
Spatial Regularity ◽  
The Right ◽  
Diffusion Convection ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Continuous Perturbation

Abstract We consider the well-posedness of a stochastic evolution problem in a bounded Lipschitz domain D ⊂ ℝ d with homogeneous Dirichlet boundary conditions and an initial condition in L2(D). The main technical difficulties in proving the result of existence and uniqueness of a solution arise from the nonlinear diffusion-convection operator in divergence form which is given by the sum of a Carathéodory function satisfying p-type growth associated with coercivity assumptions and a Lipschitz continuous perturbation. In particular, we consider the case 1 < p < 2 with an appropriate lower bound on p determined by the space dimension. Another difficulty arises from the fact that the additive stochastic perturbation with values in L2(D) on the right-hand side of the equation does not inherit the Sobolev spatial regularity from the solution as in the multiplicative noise case.

scholarly journals Well-posedness and numerical treatment of the Blackstock equation in nonlinear acoustics

2018 ◽  
Vol 28 (13) ◽  
pp. 2557-2597 ◽  
Author(s):  
Marvin Fritz ◽  
Vanja Nikolić ◽  
Barbara Wohlmuth
Keyword(s):  
Numerical Experiments ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Energy Estimates ◽  
Well Posedness ◽  
Sound Waves ◽  
Small Initial Data ◽  
Fixed Point Technique ◽  
Linearized Model ◽  
Homogeneous Dirichlet Boundary Conditions

We study the Blackstock equation which models the propagation of nonlinear sound waves through dissipative fluids. Global well-posedness of the model with homogeneous Dirichlet boundary conditions is shown for small initial data. To this end, we employ a fixed-point technique coupled with well-posedness results for a linearized model and appropriate energy estimates. Furthermore, we obtain exponential decay for the energy of the solution. We present additionally a finite element-based method for solving the Blackstock equation and illustrate the behavior of solutions through several numerical experiments.

scholarly journals The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Peter Somora
Keyword(s):  
Lower Bound ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Root Functions ◽  
Number Of Zeros ◽  
Variational Solutions ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractWe consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.

scholarly journals On the solution of two-sided fractional ordinary differential equations of Caputo type

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Ma. Elena Hernández-Hernández ◽  
Vassili N. Kolokoltsov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Well Posedness ◽  
Solutions To Equations ◽  
Fractional Differential ◽  
The Right ◽  
Stochastic Representations ◽  
Exit Problems ◽  
Special Case

AbstractThis paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes.

scholarly journals Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Anyin Xia ◽  
Xianxiang Pu ◽  
Shan Li
Keyword(s):  
Porous Medium ◽  
Global Regularity ◽  
Blow Up ◽  
Porous Medium Equation ◽  
Singularity Analysis ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Medium Equation ◽  
Regularity Results ◽  
Homogeneous Dirichlet Boundary Conditions

This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here.

scholarly journals The Evolutionary p(x)-Laplacian Equation with a Partial Boundary Value Condition

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Huashui Zhan ◽  
Zhen Zhou
Keyword(s):  
Diffusion Coefficient ◽  
Explicit Formula ◽  
Boundary Value ◽  
Electrorheological Fluids ◽  
Well Posedness ◽  
Laplacian Equation ◽  
Boundary Value Condition ◽  
Partial Boundary Value Condition ◽  
The Stability ◽  
Diffusion Convection

Consider a diffusion convection equation coming from the electrorheological fluids. If the diffusion coefficient of the equation is degenerate on the boundary, generally, we can only impose a partial boundary value condition to ensure the well-posedness of the solutions. Since the equation is nonlinear, the partial boundary value condition cannot be depicted by Fichera function. In this paper, when α<p--1, an explicit formula of the partial boundary on which we should impose the boundary value is firstly depicted. The stability of the solutions, dependent on this partial boundary value condition, is obtained. While α>p+-1, the stability of the solutions is obtained without the boundary value condition. At the same time, only if α>0 and p->1 can the uniqueness of the solutions be proved without any boundary value condition.

A Lane–Emden system with singular data

2011 ◽  
Vol 141 (6) ◽  
pp. 1279-1294 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Singular Data

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

Nonlinear Dirichlet problem with non local regional diffusion

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Dirichlet Problem ◽  
Non Local ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractThe purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problemwhere the nonlinear term

ON THE EQUATION OF BARENBLATT–SOBOLEV

2011 ◽  
Vol 13 (05) ◽  
pp. 843-862 ◽  
Author(s):  
ADIMURTHI ◽  
NGONN SEAM ◽  
GUY VALLET
Keyword(s):  
Existence And Uniqueness ◽  
Stochastic Perturbation ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Critical Value ◽  
Sobolev Equation ◽  
Lipschitz Continuous ◽  
Sobolev Problem ◽  
Uniqueness Of The Solution

In this paper, we are interested in the following pseudoparabolic problem, known as the Barenblatt–Sobolev problem: f(∂ut) - Δu - ϵΔ∂ut = g with u(0, ⋅) = u0 where f is a non-monotone Lipschitz-continuous function, ϵ > 0 and [Formula: see text]. We show the existence of a critical value ϵ0 >0 such that: if ϵ > ϵ0, then the problem admits a unique solution; if ϵ = ϵ0, the solution is unique and it exists under an additional assumption on f; if ϵ < ϵ0, then the solution is not unique in general. Passing to the limit with ϵ to 0+, we prove the existence (and uniqueness) of the solution of the Barenblatt differential inclusion Δu + g ∈ f(∂ut) for a class of maximal monotone operators f. Next, we give an extension of the main result for a stochastic perturbation of the problem and we give some numerical illustrations of the Barenblatt and the Barenblatt–Sobolev equation.

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