Sümeyye Bakım
KTO Karatay University, Engineering Faculty, Konya, Türkiye
In this talk, nonexistence of positive solutions of a fast diffusion type equation will be presented on a bounded domain $\Omega$ in a complete non compact Riemannian manifold $M$. Also new Hardy and Leray type inequalities with remainder terms on $M$ will be proved. Finally, on a model manifold hyperbolic space $\mathbb{H}^n$, several nonexistence results will be presented with the help of Hardy and Leray type inequalities.
References:
[1] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121--139.
[2] X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier, C.R. Acad.Sci. Paris, 329 (1999), 973--978.
[3] G. Carron, Inégalités de Hardy sur les variétés riemanniennes non-compactes, J. Math. Pures Appl. 76 (1997), 883--891.
[4] L. D'Ambrosio and S. Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2013), 449--475.
[5] H. Fujita, On the blowing up of solutions of the Cauchy problem for $\frac{\partial u}{\partial t}= \Delta u +u^{1+\alpha}$, J. Fac. Sci, Univ. Tokyo, Sect. 1A, Math, 13 (1966), 109--124.
[6] J. A. Goldstein and I. Kombe, Nonlinear degenerate parabolic differential equations with the singular lower order term, Advances in Differential Equations, 10 (2003), 1153--1192.
[7] I. Kombe and M. Ozaydın, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 (2009), 6191--6203.
[8] J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'hydrodynamique, J. Math. Pures et Appl. Sér. 12, (1933), 1-82.
[9] V. H. Nguyen, Sharp Hardy and Rellich type inequalities on Cartan-Hadamard manifolds and their improvements, Proc. Roy. Soc. Edinburgh Sect. A. 37 (2019), 1--30.