Elliptic regularizations for the nonlinear heat equation

Dispersive regularizations and numerical discretizations for the inviscid Burgers equation

We study centred second-order in time and space discretizations of the inviscid Burgers equation. Although this equation in its continuum formulation supports non-smooth shock wave solutions, the discrete equation generically supports smooth solitary wave solutions. Using backward error analysis we derive the modified equation associated with the numerical scheme. We identify three different eq...

Invariant Manifold of Hyperbolic-elliptic Type for Nonlinear Wave Equation

It is shown that there are plenty of hyperbolic-elliptic invariant tori, thus quasiperi-odic solutions for a class of nonlinear wave equations.

A Nonlinear Differential Equation Related to the Jacobi Elliptic Functions

A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn u,k . If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differenti...

Null Controllability of a Nonlinear Heat Equation

A blow-up result for nonlinear generalized heat equation

Available online xxxx Keywords: Nonlinear heat equation Blow up Sobolev spaces with variable exponents a b s t r a c t In this paper we consider a nonlinear heat equation with nonlinearities of variable-exponent type. We show that any solution with nontrivial initial datum blows up in finite time. We also give a two-dimension numerical example to illustrate our result.