A mollification method for a noncharacteristic Cauchy problem for a parabolic equation
The object of this paper is the study of the following noncharactristic Cauchy problem \[ u_t-a(x)u_{xx}-b(x)u_x-c(x)u=0,\quad u(0,t)=\varphi(t),\quad u_x(0,t)=0. \] The analysis of this problem is clear and was studied by the author in previous papers, too. It is proved, that under certain assumptions for the coefficients the above problem has a \(L_p\)-solution if and only if the datum \(\varphi=\varphi(t)\) belongs to a Gevrey space of order 2 basing on the \(L_p(\mathbb{R}^n)\)-norm. Moreover, the problem is ill-posed. For this reason, the author is interested in error estimates if one prescribes instead of \(\varphi\) perturbed data \(\varphi_\varepsilon\). A mollification method is proposed. The main idea is to mollify \(\varphi_\varepsilon\) in such a way that the new data belong to spaces of entire functions of exponential type. These are classes of well-posedness for noncharacteristic Cauchy problems. Some numerical examples complete the paper.
scientific article; zbMATH DE number 902580