Existence of solutions in quasi-Banach spaces for evolutionary Sobolev type equations in relatively radial case

Sobolev-type equations (equations not solved for the highest derivative) probably first appeared in the late nineteenth century. The growing recent interest in Sobolev-type equations motivates us to consider them in quasi-Banach spaces. Specifically, this study aims at understanding non-classical models of mathematical physics in quasi-Banach spaces. This paper carries over the theory of degenerate strongly continuous semigroups obtained earlier in Banach spaces to quasi-Banach spaces. We prove an analogue of the direct Hille – Yosida – Feller – Miyadera – Phillips theorem. As an application of abstract results, we consider the Showalter – Sidorov problem for modified linear Chen – Gurtin equations in quasi-Sobolev spaces.