Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results

In this work we study several questions concerning to abstract fractional Cauchy problems of order α ∈ (0, 1). Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.Conselho Nacional de Desenvolvimento Científico e TecnológicoCoordenação de aperfeiçoamento de pessoal de nivel superiorFundação de Amparo à Pesquisa do Estado de São PauloMinisterio de EducaciónMinisterio de Ciencia e InnovaciónJunta de Andalucí

Global Bifurcation of Positive Equilibria in Nonlinear Population Models

Existence of nontrivial nonnegative equilibrium solutions for age structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.Comment: 17 page