scholarly journals On non-local problems for parabolic equations

1984 ◽  
Vol 93 ◽  
pp. 109-131 ◽  
Author(s):  
J. Chabrowski
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Linear Parabolic Equation ◽  
Local Problem ◽  
Local Problems ◽  
Non Local

The main purposes of this paper are to investigate the existence and the uniqueness of a non-local problem for a linear parabolic equationin a cylinder D = Ω × (0, T].

scholarly journals A Strange Term in the Homogenization of Parabolic Equations with Two Spatial and Two Temporal Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
L. Flodén ◽  
A. Holmbom ◽  
M. Olsson Lindberg
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Time Derivative ◽  
Local Problem ◽  
Temporal Scales ◽  
Space And Time ◽  
Temporal Oscillation ◽  
Spatial Oscillations ◽  
Spatial Oscillation

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficienta(x/ε,t/ε2)in the elliptic part and spatial oscillations in the coefficientρ(x/ε)that is multiplied with the time derivative∂tuε. We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation inρ(x/ε)and the temporal oscillation ina(x/ε,t/ε2)and disappears if either of these oscillations is removed.

CAUCHY–DIRICHLET PROBLEM ASSOCIATED TO DIVERGENCE FORM PARABOLIC EQUATIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 377-393 ◽  
Author(s):  
MARIA ALESSANDRA RAGUSA
Keyword(s):  
Parabolic Equation ◽  
Dirichlet Problem ◽  
Parabolic Equations ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Second Order ◽  
Linear Parabolic Equation

In this note we study the Cauchy–Dirichlet problem related to a linear parabolic equation of second order in divergence form with discontinuous coefficients. Moreover we prove estimates in the space [Formula: see text], for every 1<p<∞.

scholarly journals Non-local Problems with Integral Displacement for Highorder Parabolic Equations

2021 ◽  
Vol 36 ◽  
pp. 14-28
Author(s):  
A.I. Kozhanov ◽  
◽  
A.V. Dyuzheva ◽  
◽  
Keyword(s):  
Parabolic Equations ◽  
High Order ◽  
Multidimensional Case ◽  
Integral Conditions ◽  
Spatial Variables ◽  
One Dimensional ◽  
Linear Parabolic Equations ◽  
Local Problems ◽  
Non Local ◽  
The One

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

scholarly journals Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on $ {\mathbb{R}}^N $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guifen Liu ◽  
Wenqiang Zhao
Keyword(s):  
Differential Equation ◽  
Parabolic Equation ◽  
White Noise ◽  
Parabolic Equations ◽  
Growth Exponent ◽  
Linear Parabolic Equation ◽  
Approximation Technique ◽  
Linear Parabolic Equations ◽  
Random Attractors ◽  
Multiplicative White Noise

<p style='text-indent:20px;'>In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of <inline-formula><tex-math id="M2">\begin{document}$ L^2( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> space. We establish the Wong-Zakai approximations of solutions in <inline-formula><tex-math id="M3">\begin{document}$ L^l( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> for arbitrary <inline-formula><tex-math id="M4">\begin{document}$ l\geq q $\end{document}</tex-math></inline-formula> in the sense of upper semi-continuity of their random attractors, where <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula> is the growth exponent of the nonlinearity. The <inline-formula><tex-math id="M6">\begin{document}$ L^l $\end{document}</tex-math></inline-formula>-pre-compactness of attractors is proved by using the truncation estimate in <inline-formula><tex-math id="M7">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and the higher-order bound of solutions.</p>

scholarly journals On some perturbation techniques for quasi-linear parabolic equations

1990 ◽  
Vol 3 (3) ◽  
pp. 169-175
Author(s):  
Igor Malyshev
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Source Function ◽  
Nonlinear Boundary ◽  
Nonlinear Operator ◽  
Nonlinear Operator Equation ◽  
Linear Parabolic Equation ◽  
Perturbation Techniques ◽  
Linear Parabolic Equations

We study a nonhomogeneous quasi-linear parabolic equation and introduce a method that allows us to find the solution of a nonlinear boundary value problem in “explicit” form. This task is accomplished by perturbing the original equation with a source function, which is then found as a solution of some nonlinear operator equation.

Existence of a Solution for a Non-Local Problem in ℝN via Bifurcation Theory

2018 ◽  
Vol 61 (3) ◽  
pp. 825-845 ◽  
Author(s):  
Claudianor O. Alves ◽  
Romildo N. de Lima ◽  
Marco A. S. Souto
Keyword(s):  
Continuous Function ◽  
Bifurcation Theory ◽  
Local Problem ◽  
Positive Continuous Function ◽  
Existence Of A Solution ◽  
Negative Function ◽  
Local Problems ◽  
Non Local ◽  
Image Position

AbstractIn this paper, we study the existence of a solution for the following class of non-local problems: P$$\eqalign{\big\{&-\Delta u=\left(\lambda f(x)-\int_{{open R}^N}K(x,y)\vert u(y)\vert ^{\gamma}\hbox{d}y\right)u\quad \mbox{in } \R^{N}, \cr &\lim_{\vert x\vert \to +\infty}u(x)=0,\quad u \gt 0 \quad \text{in } {open R}^{N},}$$ where N ≥ 3, λ > 0, γ ∈ [1, 2), f : ℝ → ℝ is a positive continuous function and K : ℝN × ℝN → ℝ is a non-negative function. The functions f and K satisfy some conditions that permit us to use bifurcation theory to prove the existence of a solution for (P).

Sign in / Sign up

Export Citation Format

Share Document