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Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form $u\sb t=Lu+Vu-\gamma u\sp p$ in $\bold R\sp n$

UC Santa Barbara Previously Published Works (2003)

The authors investigate the uniqueness and nonuniqueness of nonnegative solutions of the Cauchy problem (with nonnegative initial data f) for the second order parabolic differential equation −ut +aijDiju+biDiu+V u−u^p = 0 when the coefficients aij , bi, V , and are Holder continuous with gamma > 0 and p > 1. A key step is to prove the existence of maximal and minimal solutions of this Cauchy problem and then to derive comparison principles. Some conditions on aij and bi are given which imply the uniqueness for suitable data and some connections are given to the uniqueness (or nonuniqueness) of bounded solutions to the Cauchy problem with vanishing gamma.

$Cover page: Uniqueness/nonuniqueness for nonnegative solutions of second-order parabolic equations of the form $u\sb t=Lu+Vu-\gamma u\sp p$ in $\bold R\sp n$$