Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations

We study the strong unique continuation property backwards in time for the nonlocal equation in $\mathbb{R}^{n} \times \mathbb{R}$ \begin{equation}\label{one} (\partial_t - \Delta)^{s} u = V(x,t)u \end{equation} for $s \in (0,1)$. Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained by Poon for the case when $s=1$. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation above. With such theory in hands we establish: i) a basic monotonicity result for an adjusted frequency function which plays a central role in this paper, see Theorem 1.3; ii) an extensive blowup analysis of the so-called Almgren rescalings associated with the extension problem. We feel that our work will also be of interest e.g. in the study of certain basic open questions in free boundary problems, as well as in nonlocal segregation problems.

[1]  Arshak Petrosyan,et al.  Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem , 2013, 1306.5213.

[2]  Hui Yu.  Unique Continuation for Fractional Orders of Elliptic Equations , 2016, 1609.01376.

[3]  B. Jones.  A fundamental solution for the heat equation which is supported in a strip , 1977 .

[4]  Luis Silvestre,et al.  Chapter Four. Hölder Regularity for Generalized Master Equations with Rough Kernels , 2014 .

[5]  Angkana Ruland.  Unique Continuation for Fractional Schr\"odinger Equations with Rough Potential , 2013, 1310.7881.

[6]  J. Moser.  On a pointwise estimate for parabolic differential equations , 1971 .

[7]  Liqun Zhang,et al.  Backward uniqueness for parabolic operators with variable coefficients in a half space , 2013, 1306.3322.

[8]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[9]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[10]  Jingang Xiong,et al.  A Harnack inequality for fractional Laplace equations with lower order terms , 2010, 1101.0013.

[11]  K. Nyström,et al.  Extension properties and boundary estimates for a fractional heat operator , 2015, 1511.02893.

[12]  L. Caffarelli,et al.  On the regularity of the non-dynamic parabolic fractional obstacle problem , 2016, Journal of Differential Equations.

[13]  A. V. Balakrishnan,et al.  Fractional powers of closed operators and the semigroups generated by them. , 1960 .

[14]  J. Moser.  A Harnack inequality for parabolic di2erential equations , 1964 .

[15]  Chi-Cheung Poon.  Qnique Continuation for , 1996 .

[16]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[17]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[18]  M. Riesz.  L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .

[19]  L. Escauriaza,et al.  Unique continuation for parabolic operators , 2003 .

[20]  PABLO RAÚL STINGA,et al.  Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation , 2015, SIAM J. Math. Anal..

[21]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[22]  S. Vessella,et al.  Doubling properties of caloric functions , 2006, math/0611462.

[23]  Yannick Sire,et al.  Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates , 2010, 1012.0867.

[24]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[25]  V. M. Kenkre,et al.  Generalized master equations for continuous-time random walks , 1973 .

[26]  Marcel Riesz,et al.  Integrales de Riemann-Liouville et Potentiels , 1988 .

[27]  J. Serrin,et al.  Local behavior of solutions of quasilinear parabolic equations , 1967 .

[28]  F. Lin,et al.  Unique continuation for elliptic operators: A geometric‐variational approach , 1987 .

[29]  Kazuhiro Ishige.  On the behavior of the solutions of degenerate parabolic equations , 1999, Nagoya Mathematical Journal.

[30]  V. Felli,et al.  Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations , 2013, 1301.5119.

[31]  Aleš Nekvinda,et al.  Characterization of traces of the weighted Sobolev space $W^{1,p}(\Omega,d_M^\epsilon)$ on $M$ , 1993 .

[32]  F. Lin,et al.  Monotonicity properties of variational integrals, ap weights and unique continuation , 1986 .

[33]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[34]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[35]  N. Garofalo.  Fractional thoughts , 2017, New Developments in the Analysis of Nonlocal Operators.

[36]  R. Serapioni,et al.  A remark on a Harnack inequality for degenerate parabolic equations , 1985 .

[37]  Angkana Rüland.  Unique Continuation for Fractional Schrödinger Equations with Rough Potentials , 2013 .

[38]  Tianling Jin,et al.  On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions , 2011, 1111.1332.

[39]  Carlos E. Kenig,et al.  The local regularity of solutions of degenerate elliptic equations , 1982 .