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Pointwise bounds and blow-up for nonlinear fractional parabolic inequalities. (English. French summary) Zbl 1437.35697

Nowadays, there is a research revolution in the field of fractional calculus and analysis. There are many new definitions for fractional operators in the literature. However, the definitions of fully fractional operators have been rarely discussed in research, particularly for nonlinear Partial Differential Equations (PDEs) (refer to [P. Niu et al., Fract. Calc. Appl. Anal. 21, No. 2, 552–574 (2018; Zbl 1439.35543)]. In this research paper, the definition of fully fractional heat operator: \((\partial_{t} - \Delta)^{\alpha}: Y \rightarrow X\) such that \(X\) and \(Y\) are linear spaces, has been carefully discussed in more details with all related properties due to its important role in analyzing the initial value problem: \(0 \leq (\partial_{t} - \Delta)^{\alpha} u \leq u^{\lambda}\) in \(\operatorname{Re}^{n}\) x \(\operatorname{Re}\), \(n\geq1\), which satisfies the following initial condition: \(u=0\) in \(\operatorname{Re}^{n}\) x \((-\infty,0)\) where \(\lambda\) and \(\alpha\) are positive constants. This discussion is very helpful in studying the optimal pointwise upper bounds for the existence of nonnegative solutions, \(u(x,t)\) such that \(u \in Y\) for the nonlinear fractional parabolic inequalities from the above initial value problem when \(t \rightarrow 0^+\) and \(t \rightarrow \infty\). Novel results for the fully fractional initial value problems have been successfully established with the proof of the inverse \(J_{\alpha}\) of the fractional heat operator: \((\partial_{t} - \Delta)^{\alpha}: Y^{p}_{\alpha} \rightarrow X^{p}\) where \(\alpha,p \in \operatorname{Re}\). All results from sections \(2\) and \(4\) were fully proved in section \(6\) and \(8\), respectively. For more discussion on nonlocal and fractional modeling and analysis, we refer to [Q. Du, Nonlocal modeling, analysis, and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1423.00007)].

MSC:

35R11 Fractional partial differential equations
35B09 Positive solutions to PDEs
35B33 Critical exponents in context of PDEs
35B44 Blow-up in context of PDEs
35B45 A priori estimates in context of PDEs
35K58 Semilinear parabolic equations
35R45 Partial differential inequalities and systems of partial differential inequalities
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