Higher dimensional semilinear parabolic problems (Q1322534)

We have presented in \textit{J. W. Cholewa}, Hokkaido Math. J. 21, No. 3, 491-508 (1992)] a classical approach to the \(2m\)-th (\(m > 1\)) order initial-boundary value problem \[ \begin{cases} u_ t = -Pu + f(t,x,d^ m u) & \text{in \(D^ T = (0,T) \times G\)}\\ B_ 0 u = \dots = B_{m-1} u = 0 & \text{on \(\partial G\)}\\ u(0,x) = u_ 0(x) &\text{in \(G\)}\end{cases} \] with \(P = \sum_{| \alpha|,| \beta| \leq m}(-1)^{| \beta|} D^ \beta(a_{\alpha,\beta}(x) D^ \alpha)\), \(d^ m u = \biggl(u,{\partial u\over \partial x_ 1},\dots,{\partial u\over \partial x_ n},{\partial^ 2 u\over \partial x^ 2_ 1},\dots,{\partial^ mu\over\partial x^ m_ n}\biggr)\) and \(G \subset \mathbb{R}^ n\) a bounded domain having smooth \(C^{4m+\mu}\) boundary \(\partial G\), where \(\mu \in (0,1)\) is fixed. In this note we announce precise necessary assumptions and formulate the estimate-existence- uniqueness result in higher dimensional case \(n \geq 2m\) (especially we give the direct estimate of the ``life time'' of the solution).