An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders
The main result of the paper is a uniqueness theorem for positive solutions to the Cauchy-Dirichlet problem for a parabolic equation in a cylinder. The author considers the following problem \[ \begin{aligned} &\partial_tu=\sum_{i,j=1}^{N}\partial_j(a^{ij}(x,t)\partial_iu)+ \sum_{i=1}^{N}b^i(x,t)\partial_i u -V(x,t)u \quad \text{in } D\times (0,T),\\ &u(x,t)=0\quad \text{on }\partial D\times (0,T),\qquad u(x,0)=0\quad \text{in } D,\end{aligned}\tag{1} \] where \(D\) is an unbounded domain in \({\mathbb R}^n\) and \(T\) is a positive real number. Further, \(A(x,t)=[a^{ij}(x,t)]_{i,j=1}^{N}\) is a symmetric matrix-valued function on \( \overline{ D\times (0,T)}\) such that there exist a positive definite symmetric matrix-valued continuous function \(G(x)\) on \(\overline D\) and a positive constant \(\mu\) such that \[ \mu\langle G(x)\xi,\xi\rangle\leq \langle A(x,t)\xi,\xi\rangle\leq \mu^{-1}\langle G(x)\xi,\xi\rangle,\quad (x,t)\in\overline{ D\times (0,T)}, \quad \xi\in{\mathbb R}^N. \] The function \(G(x)\) is used to define the so-called intrinsic metric of the parabolic equation on \(D\). A certain number of hypotheses involving this metric, the growth rates of the coefficients of the parabolic equation and the domain \(D\) gives that every nonnegative solution to (1) is identically zero on \(D\times (0,T)\). The idea of defining an intrinsic metric to a parabolic equation was already used by \textit{K. Ishige} and \textit{M. Murata} [Math. Z. 227, No. 2, 313-335 (1998; Zbl 0893.35042)] and the results of the present work can be considered as improvement of those ones contained by \textit{M. Murata} [J. Funct. Anal. 135, No. 2, 456-487 (1996; Zbl 0846.35055)]. The proof is based on growth estimates of the positive solutions obtained using some Harnack type inequalities. The author discusses also the necessity of the hypotheses of his result by constructing some counter examples.
scientific article; zbMATH DE number 1384454
scientific article