RESUMEN
This paper studies a nonlinear
partial differential equation, with a discontinuous nonlinearity and a nonlocal term, which models a laser-sustained
plasma. A free boundary arises as the boundary of the plasma. It is shown that
for small values of a parameter λ, measuring laser intensity, only a
trivial solution exists. Above a critical value of λ, at least one nontrivial
solution is shown to exist. Explicit solutions are constructed which show that
multiple solutions can exist. The behaviour of the
solutions as λ → ∞ is also studied. Two types of behaviour arise; the plasma either fills all
of the bounded domain in which the problem is posed, or the Lebesgue measure of the plasma set tends to zero. In the second
case, under some assumptions on the symmetry of the domain and the coefficients
of the equation, it is shown that the plasma set is asymptotically a ball,
whose radius tends to zero at a known rate, depending on λ.Principio del formulario