Quasilinear evolutionary equations and continuous interpolation spaces

Abstract

In this paper we analyze the abstract parabolic evolutionary equations D a t ðu À xÞ þ AðuÞu ¼ f ðuÞ þ hðtÞ; uð0Þ ¼ x;in continuous interpolation spaces allowing a singularity as tk0: Here D a t denotes the timederivative of order aAð0; 2Þ: We first give a treatment of fractional derivatives in the spaces L p ðð0; TÞ; X Þ and then consider these derivatives in spaces of continuous functions having (at most) a prescribed singularity as tk0: The corresponding trace spaces are characterized and the dependence on a is demonstrated. Via maximal regularity results on the linear equationwe arrive at results on existence, uniqueness and continuation on the quasilinear equation.Finally, an example is presented. r

Convolutions, multipliers and maximal regularity on vector-valued Hardy spaces

“…(For the notion of the fractional derivative D α , see [24], §12.8, for the classical [scalarvalued] setting, or [5] for the vector-valued context.) We here consider a stronger notion of maximal regularity, in which it is also asked that u itself have the same regularity as required for Au.…”

“…Then by Lemma 6.1, the related convolution kernel and multiplier in (6.2) satisfy infinitely many of the conditions required to apply our extension results, and we obtain the boundedness of f → k * f from Theorem 5.4., or equally well the boundedness of f → F −1 [mf ] from either Theorem 5.9 or Corollary 5.11. Thus we have C 1 ⇒ C 2 , C 4 , C 5 .…”

Asymptotic Behaviors of Fundamental Solution and Its Derivatives to Fractional Diffusion-Wave Equations

“…Let α ∈ (0, 2), β ∈ (0, ∞) and p(t, x) be the fundamental solution to the space-time fractional differential equation Equation (1.1) has been an important topic in the mathematical physics related to non-Markovian diffusion processes with a memory [26,27,28,29], in the probability theory related to jump processes [5,6] and in the theory of differential equations [7,8,17,32,37]. If α ∈ (0, 1), then the fractional time derivative of order α can be used to model the anomalous diffusion exhibiting subdiffusive behavior, due to particle sticking and trapping phenomena and the fractional spatial derivative describes long range jumps of particles.…”

From particles scale to anomalous or classical convection-diffusion models with path integrals

“…We mention in particular [11] which develops global existence results of BUC solutions (even in quasilinear settings) through maximal regularity arguments for parabolic equations in the form…”

“…Occurrence of the (linear) translation operator T A −u(x)·t in 37 may be viewed as a slight perturbation of the general picture of [11]. Derivation of a priori estimates for the solutions of 37-38 is thus sufficient to ensure the existence of weak solutions.…”