Well-posedness of the Cauchy problem for nonlinear parabolic equations with variable density in the hyperbolic space
The author studies the uniqueness and nonuniqueness of bounded solutions to nonlinear Cauchy problems with variable density of the form: \[ \begin{cases} \partial _tu=\Delta _g\left[ \Phi \left( u\right) \right] ,\,& \text{in}\,\,\mathbb{H} ^n\times \left( 0,T\right] =:\mathit{H}_T, \\ u=u_0,\,& \text{in}\,\, \mathbb{H}^n\times \left\{ 0\right\}, \end{cases} \] where \(\mathbb{H}^n\) is the \(n\)-dimensional hyperbolic space \(\left( n\geq 2\right) \), \(\Delta _g\) is the Laplace-Beltrami operator in \(\mathbb{H} ^n\), \(a\) is a positive function of the space variables and denotes the density and \(u_0\) is bounded. Under some assumptions, the author shows the following: (i) Uniqueness of bounded solutions not satisfying at infinity any additional constraints, when \(a\left( x\right) \) is bounded from below by a positive constant or it goes to infinity as \(x\) tends to infinity and \(n\geq 2\). (ii) Existence of bounded solutions satisfying at infinity some extra conditions, when \(a\left( x\right) \) goes to zero as \(x\) tends to infinity and \(n\geq 2\). The structure of the paper is the following: after an introduction (Section 1), in Section 2 the author states the uniqueness and nonuniqueness results. Section 3 is dedicated to some preliminary notions of hyperbolic geometry, while in Sections 4 and 5 the author proves the uniqueness and nonuniqueness results.
scientific article; zbMATH DE number 6113122
scientific article