Mathematics Symposium, organised by the Department of Mathematical Sciences of CIÊNCIAS, featuring Leah Schätzler (Aalto University & DCM/Ciências ULisboa).
Abstract: Parabolic partial differential equations (PDEs), whose prototype is the heat equation, are usually posed in a space-time cylinder ¬ × [0, T), 0 < T , ¬ ⊂ ℝⁿ, with time 0 < T and a fixed spatial domain ¬ ⊂ ℝⁿ. However, both from a theoretical perspective and with a view to physical and biological applications, it is interesting to consider sets ¬(t) ⊂ ℝⁿ that are allowed to change in time. From the viewpoint of space-time, this means that we are dealing with a noncylindrical domain. In this talk, I will give an overview of existence and regularity results for parabolic PDEs in such noncylindrical domains. In particular, I will draw connections between the regularity of solutions and the speed at which the spatial domain is allowed to grow or shrink in time.