Semilinear Classical Wave Models

Abstract

In the previous Chaps. 18 and 19, we investigated semilinear wave models with different types of damping mechanisms. The main concern of this chapter is to give an overview on results for semilinear wave models without any damping, a never ending story in the theory of wave models. Here we distinguish between semilinear models with source and those with absorbing power nonlinearity. First we explain the Strauss conjecture for the case of source power nonlinearity. We give an overview on results for the global (in time) existence of small data solutions and for blow up behavior of solutions as well. Moreover, an overview on life span estimates completes the discussion. We explain in detail the local (in time) existence of Sobolev solutions and show for special models how Kato’s lemma is used to prove blow up of classical solutions. For wave models with absorbing power we have a more efficient energy conservation which allows for proving a well-posedness result for large data. The description of recent results on critical exponents (Strauss exponent versus Fujita exponent) for special damped wave models with power nonlinearity and on the influence of a time-dependent propagation speed in wave models on the global existence of small data weak solutions completes this chapter.