Construction of solutions of a semilinear parabolic equation with the aid of the linear Boltzmann equation
Relative to the hyperbolic first order equation on \({\mathbb{R}}^ n\times {\mathbb{R}}^+: (1)\) \(u_ t+\partial_ iA^ i(u)=0\); \(u(x,0)=u_ 0(x)\) two basically different approximation procedures are available: (i) by considering the ''viscous'' counterpart of (1), namely \((2)\quad u_ t+\partial_ iA^ i(u)=\nu \Delta u;u(x,0)=u_ 0(x);\nu >0\) (ii) by considering solutions of the linear Boltzmann equation \((3)\quad f_ t+\partial_ if\cdot \xi^ i=0on{\mathbb{R}}^ n\times {\mathbb{R}}^+;f(x,\xi,0)=f_ 0(x,\xi),\) where \(\xi \in {\mathbb{R}}^ n\) can be thought of as a parameter. In this case, once a certain relationship holds between the initial datum of (3) and that of (1), involving smoothing with an approximate of the \(\delta\)-measure, \(\chi_{\epsilon}\), it is known that for fixed \(\epsilon\) a certain iterate of the (\(\xi\)-averaged) solution of (3) converges to the solution of (1). The parameter in terms of which convergence takes place can be viewed as a stepsize h, \(h\to 0.\) The present paper deals with the case in which h still converges to zero, but the smoothing parameter \(\epsilon\), instead of being fixed, goes to zero, too, under the assumption that \(h/\epsilon^ 2\) is a fixed constant. The author's main result is that in this case the limit exists, too, and it is precisely the solution of (2) with \(\nu\) determined by the ratio \(h/\epsilon^ 2\). This nice result, which sheds light on the role of linear diffusion models, is proved in a careful way relying on an approximation result for nonlinear semigroups by \textit{S. Oharu} and \textit{T. Takahashi} [J. Math. Soc. Japan 26, 124-160 (1974; Zbl 0265.47052)].
semilinear first order hyperbolic equations
scientific article; zbMATH DE number 3895522