The Critical Exponent(s) for the Semilinear Fractional Diffusive Equation

Author(s)

D'Abbicco, Marcello ^[1] ; Ebert, Marcelo Rempel ^[2] ; Picon, Tiago Henrique ^[2]

Affiliation

^[1] Univ Bari, Dept Math, Via E Orabona 4, I-70125 Bari - Italy

^[2] Univ Sao Paulo, FFCLRP, Dept Comp & Matemat, Ave Bandeirantes 3900, BR-14040901 Ribeirao Preto, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS; v. 25, n. 3, p. 696-731, JUN 2019.
• Web of Science Citations: 1

Abstract

In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation [partial derivative(1+alpha)(t)u - Delta u = vertical bar u vertical bar(p), t >= 0, x is an element of R-n, u(0, x) = u(0)(x), x is an element of R-n, u(t)(0, x) = u(1()x) x is an element of R-n, where alpha is an element of( 0, 1), and partial derivative(1+alpha)(t)u is the Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional, and so the role played by the second data u1 becomes ``unnatural{''} as alpha decreases to zero. To prove our result, we first derive L-r - L-q estimates, 1 <= r <= q <= infinity, for the solution to the linear Cauchy problem, where u vertical bar(p) is replaced by f (t, x), and then we apply a contraction argument. (AU)

• FAPESP's process: 13/17636-5 - A priori estimates for elliptic complexes and applications
• Grantee: Tiago Henrique Picon
• Support Opportunities: Research Grants - Young Investigators Grants
• FAPESP's process: 15/16038-2 - Lp-Lq decay estimates for Evolution Operators
• Grantee: Marcelo Rempel Ebert
• Support Opportunities: Regular Research Grants