In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation $u_t = \Delta u + u^p $ in $R^N $ with $p > 1$ and nonnegative initial values. Fujita showed that if $1 1 + {2 / N}$, there were nontrivial global solutions. This paper discusses similar results for other geometries and other equations including a nonlinear wave equation and a nonlinear Schrodinger equation.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1048&context=math_pubs

The role of critical exponents in blowup theorems

Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data

We prove that, if the initial data has moments inv higher than three, then the solution of Vlasov-Poisson has also moments inv higher than three. We deduce from this different regularity results on the local density, the force field or the solution itself. Also we give a new uniqueness result, and new regularity results for solutions satisfying only the energy andL
∞ bounds. Our proofs are based on a new representation formula and logarithmic estimates for the force field.

Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

Recently Plaffelmoser has shown that solutions of the Vlasov–Poisson system in three dimensions remain smooth for all time. This paper establish the same existence theorem by a simpler method

Global existence of smooth solutions to the vlasov poisson system in three dimensions

In a previous publication the authors have obtained time decay rates for solutions to the initial-value problem for the linearized Vlasov equation (in one space dimension).

On time decay rates in landau damping

Continuous finite element methods which preserve energy properties for nonlinear problems

On considere l'existence de solutions classiques globales aux problemes de Cauchy de la forme U tt -△ u =H(u x ,u t ) sur (x,t)∈R n ×(o,∞) u(x,o)=f(x) sur x∈R n , u t (x,o)=g(x) sur x∈R n pour des donnees de Cauchy petites

Finite-Time Blow-Up for in Two Space Dimensions

By minimizing the energy for the Vlasov-Poisson system under a constraint, Guo and Rein have constructed a large class of isotropic, spherically symmetric steady states. They have shown that an isolated minimizer is automatically dynamically stable under general (i.e., not necessarily symmetric) perturbations. The main result of this work is to remove the assumption that the minimizer must be isolated, so minimizers are stable even if they are not isolated. It is also shown that the Lagrange multipliers associated with all minimizers have the same value. Finally, an example where two distinct minimizers exist is studied numerically.

Jack Schaeffer

Papers

On time decay rates in landau damping

Continuous finite element methods which preserve energy properties for nonlinear problems

Finite-Time Blow-Up for in Two Space Dimensions

Steady States in Galactic Dynamics

Global existence for the Poisson-Vlasov system with nearly symmetric data