Stability results for a class of non-linear parabolic equations

Summary We study the asymptotic behaviour of the solutions of the equation u_t=Au+λu−|u|^αu. Denoting by λ₀ the principal eigenvalue of the second-order differential operator A, we shall prove that if λ ⩽ λ₀ the only equilibrium solution, namely zero, is asymptotically stable, whereas, if λ>λ₀, the nontrivial equilibrium solutions without internal zeros are asymptotically stable. Attractivity and stability are proved both in the L²-norm and in the H ₀ ¹ -norm. Entrata in Redazione il 15 ottobre 1976.     

Quasilinear resolutions of non-linear equations

By analogy with the linear vector bundle case, a non-linear partial differential equation on a manifold can be defined as a fibred submanifold R_k of a k-jet bundle. By observing that under natural conditions the first prolongation gives rise to a vector bundle over R_k, (that is, a quasilinear equation), techniques of the linear case are adapted to establish conditions for the formal integrability of the equation.     

Numerical solution of non-linear Volterra integral equations

This paper deals with non-linear Volterra integral equations of the type $\text{y(x)}\text{=}\text{f(x)}\text{+ ?}{}_0{}^{\mathrm{<mtext>x</mtext>}}\text{H[t, x, y (t), y (x)] dt}$. Convergence criteria are given (in the same sense of the maximum and C^a norms) for the numerical solution of this type of Volterra integral equation. Several numerical methods are compared.     

Numerical solutions of two non-linear simultaneous equations

In this article, we shall give practical and numerical representations of inverse mappings of two-dimensional mappings (of the solutions of two non-linear simultaneous equations) and show their numerical experiments by using computers. We derive a concrete formula from a very general idea for the representation of the inverse functions.     

On a system of non-linear differential equations

Summary E. Kasner(1925) considered a system of non-linear differential equations in three variables. The authors of this paper have extended this system to n variables by means of a linear transformation with complex coefficients. The formulas obtained for the solutions of the differential equations give a simplification of Kasner’s solutions. Entrata in Redazione il 26 agosto 1968.     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

Linearly implicit time discretization of non-linear parabolic equations

We give a stability and error analysis of linearly implicitone-step methods for time discretization of non-linear parabolicequations. We derive precise error bounds for Rosenbrock andW-methods, and we explain the error reduction by Richardsonextrapolation of the linearly implicit Euler method which occursin spite of the breakdown of asymptotic expansions. The parabolicequations are studied in a Hilbert space framework that includessemilinear and quasilinear parabolic equations, and also stiffreaction-diffusion equations with reactions at different timescales.