Initial-boundary value problems for conservative Kimura-type equations: solvability, asymptotic and conservation law

We consider the linear degenerate parabolic equation ∂u∂t-xa0(x,t)∂2u∂x2+a1(x,t)∂u∂x+a2(x,t)u=f(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial u}{\partial t}-x a_{0}(x,t)\frac{\partial ^{2}u}{\partial x^{2}}+a_1(x,t)\frac{\partial u}{\partial x}+a_2(x,t)u=f(x,t) \end{aligned}$$\end{document}originated from pandemic dynamics modeling. Under suitable conditions on the given data, the global classical solvability to the related initial-boundary value problem is addressed without a prescribing boundary condition at the origin. Also, we show that under some assumptions on regularity of coefficients and initial data, classical solutions vanish at the origin on any finite time interval. Besides, we establish that vanishing at the origin of solutions is consistent with the conservation property of the model.