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We prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary conditions. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies on a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order. The potentials include all the nonnegative ones. For the first two equations, we prove if satisfies some growth conditions in , then is analytic in time. Here is or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that is analytic in time at. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general. For the nonlinear heat equation with power nonlinearity of order , we prove that a solution is analytic in time if it is bounded in and is a positive integer. In addition, we investigate the case when is a rational number with a stronger assumption. It is also shown that a solution may not be analytic in time if it is allowed to be. As a lemma, we obtain an estimate of where is the heat kernel on a manifold, with an explicit estimation of the coefficients. An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable , implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives. [ABSTRACT FROM AUTHOR]

In this paper we consider the Cauchy problem for the heat equation with a nonnegative potential decaying quadratically at the space infinity and investigate local concavity properties of the solution. In particular, we give a sufficient condition for the solution to be quasi-concave in a ball for any sufficiently large t, and discuss the optimality of the sufficient condition, identifying a threshold for the occurrence of local quasi-concavity. [ABSTRACT FROM AUTHOR]