The zero set of a solution of a parabolic equation. (English) Zbl 0644.35050

Under very mild conditions on the coefficients a, b and c it is shown that for any fixed time \(0<t<T\) a solution u(x,t) of \[ (1)\quad u_ t=a(x,t)u_{xx}+b(x,t)u_ x+c(x,t)u\quad (x\in {\mathbb{R}},\quad 0<t<T) \] has a locally finite number of zeroes in the x-direction. The solution is assumed to satisfy an estimate of the type \(| u(x,t)| \leq A \exp (Bx^ 2)\) for some \(A,B>0.\)
Furthermore solutions of (1) on finite intervals are studied. In this case it is shown that the number of zeroes does not increase with time, and actually decreases whenever u(t,\(\cdot)\) has a degenerate zero.
Reviewer: S.B.Angenent


35K10 Second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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