Continuous dependence of nonnegative solutions of the heat equation on noncharacteristic cauchy data

Abstract

The sideways Caucht problem for the one-dimensional heat, in which the solution and its first-order spatial derivative are specified on an interval ofthe time axis, is well known to be ill-osed. Newvertheless, we establish continuous dependence on th Cauchy data of nonnegative solutions satisfying the extra smoothness requriement that the time derivativer is continuous on the initial manifold. This is done by first establishing explicit boubnds for suchg solutions and certain of their derivatives and then applying result of J. Cannon. Most of the estimates, which may be of interest in themselves, do not depend on the extra smoothness assumption. In spite of the fact that the admissible Cauchy data is hightly non-arbitrary, our results include the fact that the set of admissible Cauchy data is closed under unifrom converence of the data and one first-order derivative of the data.

Keywords:

• heat equation
• ill-posed Cauchy problem

AMS:

• 35B30
• 35B45
• 35K05
• 35R25