A Free Boundary Problem in Plasma Containment

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$\lambda $, measuring laser intensity, only a trivial solution exists. Above a critical value of $\lambda $, 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 $\lambda \to \infty $ 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 $\lambda $.

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