2015/05/11 by Jukka Kemppainen, Kemppainen, Jukka, Juhana Siljander +3 · 4 citations
Engineering · Mathematics · #35C15 #47G20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary 35R11 #Secondary 45K05 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1505.02803
openalex publication_date 2015/05/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We study the Cauchy problem for a nonlocal heat equation, which is of\nfractional order both in space and time. We prove four main theorems:\n (i) a representation formula for classical solutions,\n (ii) a quantitative decay rate at which the solution tends to the fundamental\nsolution,\n (iii) optimal L2-decay of mild solutions in all dimensions,\n (iv) L2-decay of weak solutions via energy methods.\n The first result relies on a delicate analysis of the definition of classical\nsolutions. After proving the representation formula we carefully analyze the\nintegral representation to obtain the quantitative decay rates of (ii).\n Next we use Fourier analysis techniques to obtain the optimal decay rate for\nmild solutions. Here we encounter the critical dimension phenomenon where the\ndecay rate attains the decay rate of that in a bounded domain for large enough\ndimensions. Consequently, the decay rate does not anymore improve when the\ndimension increases. The theory is markedly different from that of the standard\ncaloric functions and this substantially complicates the analysis.\n Finally, we use energy estimates and a comparison principle to prove a\nquantitative decay rate for weak solutions defined via a variational\nformulation. Our main idea is to show that the L2-norm is actually a\nsubsolution to a purely time-fractional problem which allows us to use the\nknown theory to obtain the result.\n