On the large time behaviour of solutions to the Dirichlet problem for a parabolic equation with singular coefficients (Q1382572)

Let \(\Omega\) be bounded domain in \(\mathbb{R}^n\) with piecewise smooth boundary \(\partial\Omega\), \(G = \Omega\times\{t>0\}\). Considered is the equation \[ \dfrac{\partial u}{\partial t} - \sum_{i,j=1}^n a_{ij}(x,t) \dfrac{\partial^2 u}{\partial x_i\partial x_j} - \sum_{i=1}^n b_i(x,t) \dfrac{\partial u}{\partial x_i} - c(x,t)u = 0\tag{1} \] for \((x,t)\in G\), where \(a_{ij}\equiv a_{ji}\), \(u(x,t)= 0\) for \(x\in\partial\Omega\), \(t>0\), \(u(x,0)= \varphi(x)\) for \(x\in\overline\Omega\), \(\varphi\in C(\overline\Omega)\), \(\varphi(x)= 0\) for \(x\in\partial\Omega\). The author establishes conditions on the coefficients of equation (1) for which the solution \(u(x,t)\) exponentially decreases for any continuous initial function.