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

The motion of a collisionless plasma is modeled by solutions to the Vlasov–Maxwell system. The Cauchy problem for the relativistic Vlasov–Maxwell system is studied in the case when the phase space distribution function f = f(t,x,v) depends on the time t,  and . Global existence of classical solutions is obtained for smooth data of unrestricted size. A sufficient condition for global smooth solvability is known from [12]: smooth solutions can break down only if particles of the plasma approach the speed of light. An a priori bound is obtained on the velocity support of the distribution function, from which the result follows.

The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

Global classical solutions to the initial value problem for the relativistic Vlasov-Maxwell equations are obtained in three space dimensions. The initial distribution of the various species may be large, provided that the total positive charge nearly cancels the total negative charge.

/pdf/global-existence-for-the-relativistic-vlasov-maxwell-system-3rwseinqgw.pdf

Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data

Spherically symmetric solutions to the Cauchy problem for the relativistic Vlasov-Poisson system are studied in three space dimensions. If the energy is positive definite (the plasma physics case), global classical solutions exist. In the case of indefinite energy, “small” radial solutions exist in the large, but “large” data solutions (those with negative energy) will blow-up in finite time.

/pdf/on-symmetric-solutions-of-the-relativistic-vlasov-poisson-16npocxkws.pdf

On symmetric solutions of the relativistic Vlasov-Poisson system

Solutions of the relativistic Vlasov-Maxwell system of partial differential equations are considered in three space dimensions. The speed of light,c, appears as a parameter in this system. For smooth Cauchy data, classical solutions are shown to exist on a time interval that is independent ofc. Then, using an integral representation for the electric and magnetic fields due to Glassey and Strauss [6], conditions are given under which solutions of the relativistic Vlasov-Maxwell system converge in pointwise sense to solutions of the non-relativistic Vlasov-Poisson system at the asymptotic rate of 1/c, asc tends to infinity.

/pdf/the-classical-limit-of-the-relativistic-vlasov-maxwell-iuoudvj2sh.pdf

Jack Schaeffer

Papers

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

The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data

On symmetric solutions of the relativistic Vlasov-Poisson system

The classical limit of the relativistic Vlasov-Maxwell system