A semilinear problem associated to the space-time fractional heat equation in RN

• Published in: Calculus of variations and partial differential equations 2024, Vol.63 (9)
• Main Authors: Cortázar, Carmen, Quirós, Fernando, Wolanski, Noemí
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; Thermodynamics
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-024-02836-z

We study the fully nonlocal semilinear equation ∂ t α u + ( - Δ ) β u = | u | p - 1 u , p ⩾ 1 , where ∂ t α stands for the usual time derivative when α = 1 and for the Caputo α -derivative if α ∈ ( 0 , 1 ) , while ( - Δ ) β , β ∈ ( 0 , 1 ] , is the usual β power of the Laplacian. We prescribe an initial datum in L q ( R N ) . We give conditions ensuring the existence and uniqueness of a solution living in L q ( R N ) up to a maximal existence time T that may be finite or infinite. If  T is finite, the L q norm of the solution becomes unbounded as time approaches T , and u is said to blow up in L q . Otherwise, the solution is global in time. For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. Our weakest condition for global existence and our condition for blow-up are both related to the size of the averages of the initial datum in balls. As a corollary, every nonnegative nontrivial solution in L q blows up in finite time if 1 < p < p f : = 1 + 2 β N whereas if p > p f there are both solutions that blow up and global ones. Noteworthy, the critical Fujita-type exponent p f does not depend on α . However, there is an important difference in the behavior of solutions in the critical case p = p f depending on the value of this parameter: when α = 1 it was known that all nonnegative and nontrivial solutions blow up, while we prove here that if α ∈ ( 0 , 1 ) there is global existence for some initial data.