scholarly journals Explicit Blowing Up Solutions for a Higher Order Parabolic Equation with Hessian Nonlinearity

2021 ◽  
Author(s):  
Carlos Escudero
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Higher Order ◽  
Order Partial Differential Equation ◽  
Infinite Time ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Blowing Up ◽  
Blowing Up Solutions

AbstractIn this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.

scholarly journals Blow-up criterion for the density dependent inviscid Boussinesq equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Li ◽  
Yanping Zhou
Keyword(s):  
Velocity Field ◽  
Energy Method ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Smooth Solutions ◽  
Density Dependent ◽  
Priori Estimates ◽  
Blow Up Criterion

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in $\mathbb{R}^{N}\ (N\geq 2)$ R N ( N ≥ 2 ) . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

scholarly journals Positive solutions of higher order quasilinear elliptic equations

2002 ◽  
Vol 7 (8) ◽  
pp. 423-452
Author(s):  
Marcelo Montenegro
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
A Priori Estimate ◽  
Higher Order ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Radial Solutions ◽  
The Maximum Principle ◽  
Priori Estimates ◽  
Quasilinear Elliptic

The higher order quasilinear elliptic equation−Δ(Δp(Δu))=f(x,u)subject to Dirichlet boundary conditions may have unique and regular positive solution. If the domain is a ball, we obtain a priori estimate to the radial solutions via blowup. Extensions to systems and general domains are also presented. The basic ingredients are the maximum principle, Moser iterative scheme, an eigenvalue problem, a priori estimates by rescalings, sub/supersolutions, and Krasnosel'skiĭ fixed point theorem.

scholarly journals A priori estimates for higher order multipliers on a circle

2002 ◽  
Vol 130 (10) ◽  
pp. 3043-3050 ◽  
Author(s):  
A. Alexandrou Himonas ◽  
Gerard Misiołek
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Higher Order ◽  
Priori Estimates

A nonlinear partial differential system arising in thermoelectricity

2005 ◽  
Vol 16 (6) ◽  
pp. 683-712 ◽  
Author(s):  
A. BERMÚDEZ ◽  
R. MUÑOZ-SOLA ◽  
F. PENA
Keyword(s):  
Source Term ◽  
A Priori Estimates ◽  
A Priori ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Existence Of A Solution ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Partial Differential System ◽  
Temperature Equation

In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

scholarly journals On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities

2000 ◽  
Vol 160 ◽  
pp. 123-134 ◽  
Author(s):  
Haruo Nagase
Keyword(s):  
Continuous Dependence ◽  
A Priori Estimates ◽  
Lower Semicontinuous ◽  
A Priori ◽  
Regularity Property ◽  
Regularity Properties ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Lower Semicontinuous Functional ◽  
Continuous Dependence Of Solutions

AbstractIn this paper we consider the following nonlinear parabolic variational inequality; u(t) ∈ D(Φ) for all where Δp is the so-called p-Laplace operator and Φ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data u0 and f.

A Priori Estimates for Solutions of "Sub-critical" Equations on CR Sphere

2003 ◽  
Vol 3 (3) ◽  
Author(s):  
J. Prajapat ◽  
Mythily Ramaswamy
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Finite Energy ◽  
Priori Estimates

AbstractHere we study the precise blow-up behaviour and obtain a priori estimates for the finite energy Con the odd dimensional spheres S

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