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SUMMARY:Strong dissipativity of generalized time-fractional derivatives an
 d quasi-linear (stochastic) partial differential equations - Michael Roeck
 ner (Universität Bielefeld)
DTSTART:20220125T100000Z
DTEND:20220125T103000Z
UID:TALK169157@talks.cam.ac.uk
DESCRIPTION:3) Michael R&ouml\;ckner (Bielefeld University and Academy of 
 Mathematics and Systems Science\, Chinese Academy of Sciences\, Beijing)\n
 Joint work with Wei Liu (Jiangsu Normal University\, Xuzhou) and Jos\\'e L
 u\\'is da Silva (University of Madeira\, Funchal)\nTitle: Strong dissipati
 vity of generalized time-fractional derivatives and quasi-linear (stochast
 ic) partial differential equations\nAbstract:\nIn this talk we shall ident
 ify generalized time-fractional derivatives as generators of $C_0$-operato
 r semigroups and prove their strong dissipativity on Gelfand triples of pr
 operly in time weighted $L^2$-path spaces. In particular\, the classical C
 aputo derivative is included as a special case. As a consequence\, one obt
 ains the existence and uniqueness of solutions to evolution equations on G
 elfand triples with generalized time-fractional derivatives. These equatio
 ns are of type\n\\begin{equation*}\n\\frac{d}{dt} (k * u)(t) + A(t\, u(t))
  = f(t)\, \\quad 0<t<T\,\n\\end{equation*}\nwith (in general nonlinear) op
 erators $A(t\,\\cdot)$ satisfying general weak monotonicity conditions. He
 re $k$ is a non-increasing locally Lebesgue-integrable nonnegative functio
 n on $[0\, \\infty)$ with $\\underset{s\\rightarrow\\infty}{\\lim}k(s)=0$.
 \nAnalogous results for the case\, where $f$ is replaced by a time-fractio
 nal additive noise\, are obtained as well. Applications include generalize
 d time-fractional quasi-linear (stochastic) partial differential equations
 . In particular\, time-fractional (stochastic) porous medium and fast diff
 usion equations with ordinary or fractional Laplace operators or the time-
 fractional (stochastic) $p$-Laplace equation are covered.\nWork supported 
 by Deutsche Forschungsgemeinschaft (DFG) through ``Collaborative Research 
 Centre (CRC) 1283''. References: Liu/R./Silva: arXiv:1908.03959
LOCATION:Seminar Room 2\, Newton Institute
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