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On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients. (English) Zbl 0869.35010

Let \(X\) be a domain in \(\mathbb{R}^n\) and consider a second-order parabolic operator \(Lu=u_t+\rho(x)Pu\) defined on \(X\times[0,T)\). Here \(\rho\) is a positive function and \(P\) is a time independent elliptic operator of the form \[ P(x,\partial_x)=- \sum^n_{i,j=1} a_{ij}(x)\partial_i\partial_j+ \sum^n_{i=1} b_i(x)\partial_i+ c(x). \] A nonnegative solution \(u(x,t)\) of the equation \(Lu=0\) in \(X\times [0,T)\) with nonnegative initial data \(u(x,0)= u_0(x)\) is called a solution of the positive Cauchy problem.
We study the uniqueness (UP) and nonuniqueness (NUP) of the positive Cauchy problem. In the first part of the paper, we show that for two classes of parabolic operators nonnegative solutions of the equation \(Lu=0\) satisfy a (restricted) uniform Harnack inequality, a condition which implies UP. The second part of the paper is devoted to nonuniqueness results. Using a general sufficient condition, we prove (NUP) for a class of radially symmetric Schrödinger semigroups in \(\mathbb{R}^n\).

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K15 Initial value problems for second-order parabolic equations
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