The authors prove nonexistence of global solutions in time of the initial boundary value problem
where , , and is a bounded domain of , with a smooth boundary . This complicated equation is said to appear in nonlinear viscoelasticity theory. The authors suppose that there exist solutions in a function space , where
for . Local existence and uniqueness are not especially discussed. The energy is defined by
Let denote the Sobolev critical exponent of . There is just one theorem.
Theorem: Assume that and . If , then the solution cannot exist for all time.
The authors insist that their result improves one by Z. Yang [Math. Methods Appl. Sci. 25, No. 10, 825–833 (2002; Zbl 1009.35051)], whose result requires an additional assumption, (a constant depending on the size of ).
The proof is performed as follows. Define
where . The main aim of the proof is to show that for suitable ,
It follows easily from this inequality that attains in finite time. The mathematical tool of higher grade are not needed, but silful calculation is performed to obtain the inequality. Young’s inequality, and Sobolev’s inequality, Poincarés inequaly in bounded domains are exploited.
Global existence is not especially referred.