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Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications. (English. Russian original) Zbl 0582.35052

Russ. Math. Surv. 39, No. 3, 119-178 (1984); translation from Usp. Mat. Nauk 39, No. 3(237), 107-156 (1984).
The following parabolic equation is considered: \[ \partial_ t u- \{a_{ij}(t,x) \partial_{x_ i} \partial_{x_ j} u+b_ i(t,x) \partial_{x_ i} u+c(t,x)u\}=0 \] whose coefficients \(a_{ij}\), \(b_ i\) and c are continuously bounded and satisfy uniformly (with respect to t) Dini’s continuity condition in x. Also there exists \(\mu\geq 1\) such that \[ \mu^{-1} | \xi |^ 2\leq a_{ij}(t,x) \xi_ i \xi_ j\leq \mu | \xi |^ 2,\quad \forall \xi \in \mathbb R^ n. \] For this equation and others obtained by specializing the coefficients and related conditions there are proved different results such as: existence and properties for the solutions of the Cauchy problem associated with it, Harnack’s inequality and its implications to the properties of the solutions, two-sided estimates of the fundamental solutions, asymptotic behaviour (for \(t\to \infty)\) of the solutions etc. Also elliptic and parabolic equations with measurable coefficients are considered; the properties of the weak fundamental solutions being established via the energy inequality. The paper ends with a very useful section of comments.

MSC:

35K15 Initial value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
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