scholarly journals ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

2021 ◽  
Author(s):  
Iasson Karafyllis ◽  
Miroslav Krstic
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Lyapunov Functional ◽  
The State ◽  
Classical Solutions ◽  
Parabolic Partial Differential Equations ◽  
Maximum Principles ◽  
Fading Memory ◽  
Parabolic Pdes ◽  
Non Local

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.

scholarly journals Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

2019 ◽  
Vol 150 (1) ◽  
pp. 475-495 ◽  
Author(s):  
Begoña Barrios ◽  
Maria Medina
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Classical Case ◽  
Parabolic Problems ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Local Version ◽  
Comparison Results ◽  
Non Local

AbstractWe present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical cases= 1 in [23, 24] respectively.

scholarly journals Global Bounded Classical Solutions for a Gradient-Driven Mathematical Model of Antiangiogenesis in Tumor Growth

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Xiaofei Yang ◽  
Bo Lu
Keyword(s):  
Mathematical Model ◽  
Endothelial Cells ◽  
Partial Differential Equations ◽  
Global Existence ◽  
Tumor Growth ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Angiogenic Factors ◽  
Classical Solutions ◽  
Parabolic Partial Differential Equations

In this paper, we consider a gradient-driven mathematical model of antiangiogenesis in tumor growth. In the model, the movement of endothelial cells is governed by diffusion of themselves and chemotaxis in response to gradients of tumor angiogenic factors and angiostatin. The concentration of tumor angiogenic factors and angiostatin is assumed to diffuse and decay. The resulting system consists of three parabolic partial differential equations. In the present paper, we study the global existence and boundedness of classical solutions of the system under homogeneous Neumann boundary conditions.

SOME MAXIMUM PRINCIPLES AND SYMMETRY RESULTS FOR A CLASS OF BOUNDARY VALUE PROBLEMS INVOLVING THE MONGE-AMPÈRE EQUATION

2001 ◽  
Vol 11 (06) ◽  
pp. 1073-1080 ◽  
Author(s):  
G. A. PHILIPPIN ◽  
A. SAFOUI
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Recent Result ◽  
Boundary Value Problems ◽  
Differential Inequality ◽  
Boundary Value ◽  
Maximum Principles ◽  
Maximum Value ◽  
Symmetry Properties ◽  
Monge Ampere

In this paper we investigate a class of boundary value problems for the Monge-Ampère equation [Formula: see text] where Ω is a strictly convex bounded domain in RN, N≥2. When f=g(u)h(|∇u|2) with g and h satisfying the differential inequality [Formula: see text] we show in Sec. 2 that the function [Formula: see text] takes its maximum value on the boundary ∂Ω. This maximum principle generalizes a recent result of Ma who investigated the case f= const in R2. In Sec. 3 we investigate symmetry properties of u under specific boundary conditions or geometry of Ω.

scholarly journals MAXIMUM PRINCIPLES AROUND AN EIGENVALUE WITH CONSTANT EIGENFUNCTIONS

2008 ◽  
Vol 10 (06) ◽  
pp. 1243-1259 ◽  
Author(s):  
J. CAMPOS ◽  
J. MAWHIN ◽  
R. ORTEGA
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Neumann Boundary ◽  
Function Spaces ◽  
Strong Maximum Principle ◽  
Neumann Boundary Conditions ◽  
Linear Operators ◽  
Maximum Principles ◽  
Partial Differential Operators

A class of linear operators L + λI between suitable function spaces is considered, when 0 is an eigenvalue of L with constant eigenfunctions. It is proved that L + λI satisfies a strong maximum principle when λ belongs to a suitable pointed left-neighborhood of 0, and satisfies a strong uniform anti-maximum principle when λ belongs to a suitable pointed right-neighborhood of 0. Applications are given to various types of ordinary or partial differential operators with periodic or Neumann boundary conditions.

Elliptic problem involving non-local boundary conditions

2019 ◽  
Vol 181 ◽  
pp. 87-100
Author(s):  
Noureddine Igbida ◽  
Soma Safimba
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problem ◽  
Local Boundary ◽  
Non Local
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