## 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

### MSC:

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

### Keywords:

locally finite; number of zeroes
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