Abstract. We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems:(i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal L 2 -decay of mild solutions in all dimensions, (iv) L 2 -decay of weak solutions via energy methods.The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii).Next we use Fourier analysis techniques to obtain the optimal decay rate for mild solutions. Here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions. Consequently, the decay rate does not anymore improve when the dimension increases. The theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis.Finally, we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation. Our main idea is to show that the L 2 -norm is actually a subsolution to a purely time-fractional problem which allows us to use the known theory to obtain the result.

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in R d . An important special case is the timefractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young's inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a critical dimension phenomenon. The general subdiffusion case is treated by method (B) and relies on a careful estimation of the underlying relaxation function. Several examples of kernels, including the ultraslow diffusion case, illustrate our results.

We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods. 2000 Mathematics Subject Classification. Primary 35B65. Secondary 35K65, 35D10. Key words and phrases. Hölder continuity, Caccioppoli estimates, intrinsic scaling, Harnack's inequality. Research of JMU supported by CMUC/FCT and project UTAustin/MAT/0035/2008. 1 2 KUUSI, SILJANDER AND URBANO two complementary cases: Case I : 0 ≤ ess inf u << ess osc u and Case II : u p−2 u t ≈ Cu t .In large scales, i.e., in Case I, the scaling property of the equation dominates and, consequently, the reduction of the oscillation follows immediately from Harnack's inequality. In small scales, on the other hand, the oscillation is already very small and thus the solution itself is between two constants, the infimum and the supremum, whose difference is negligible. Correspondingly, the nonlinear time derivative term, which formally looks like u p−2 u t , behaves like u t and we end up with a p-parabolic type behavior. However, also in this case, we still need to modify the known arguments. In particular, the energy estimates are not available in the usual form and we need to use modified versions as in [5], [15] and [25].Our argument also applies to doubly nonlinear equations of p-Laplacian type that are of the formwith A(x, t, ·, ·) satisfying the usual structure assumptions. For expository purposes, we only consider (1.1). Very recently, a direct geometric method to obtain local Hölder continuity for parabolic equations has been developed in [6] and [9]. Despite the effort, the general picture remains unclear. Preliminaries

Abstract. We complete the study of the regularity for Trudinger's equation by proving that weak solutions are Hölder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.