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This is certainly possible, and it seems rather obvious on physical grounds. Consider an initial condition where you have two hot regions connected by a thin corridor which is also hot, and the surrou …

Isn't the Laplacian with Neumann conditions a counterexample? There are no lower order terms, yet there is no inverse.

Under reasonable boundary conditions, you will have $$\|D_3f\|_{L^2}\le C\|f\|_{H^3}\le C\|f\|_{H^4}^{1/2}\|f\|_{H^2}^{1/2}\le C\|\Delta^2f\|_{L^2}^{1/2}\|\Delta f\|_{L^2}^{1/2}\le C\lambda^{3/2}\|f\| …

The standard references on parabolic PDEs assume bounded coefficients. When dealing with Fokker-Planck equations, however, the coefficients are usually unbounded. The way to get around this is to appr …

Example for first question: $\Delta^2 u=0$, say on the unit disk, with boundary condition ${\partial\over\partial n}\Delta u=0$, $\Delta u-u=0$. The boundary conditions do not satisfy the Lopatinskii …

No. Consider the case $p=0$. In that case, solutions are constant along characteristics. But polynomial growth does not preclude characteristics from diverging to infinity in finite time.

The answer is yes. Suppose $g$ is orthogonal to the image of $S$, and let $v$ be the solution of the Dirichlet problem $\Delta v=g\delta(\partial D_1)$ on $D_2$, where $\delta(\partial D_1)$ is a delt …

The question whether appropriate weights exist is discussed in the following paper: L.R. Volevich, A problem of linear programming arising in differential equations, Uspekhi Mat. Nauk 18 (1963), No. 3 …