Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations

Strichartz estimates and smooth attractors of dissipative hyperbolic equations

In this thesis we prove attractor existence and its smoothness for several classes of damped wave equations with critical nonlinearity. The term " critical " refers to the fact that behaviour of the solutions is determined not only by the energy but also by some more subtle space-time norms which are known as Strichartz norms. One of the main achievements of the work is the construction of the ...

Solutions of Nonlinear Hyperbolic Equations at Resonance

Solutions of Smooth Nonlinear Partial Differential Equations

and Applied Analysis 3 definition, it may happen that such a solution does not belong to any of the customary spaces of generalized functions. For example, given a function u : C \ {z0} −→ C 1.3 which is analytic everywhere except at the single point z0 ∈ C, and with an essential singularity at z0, Picard’s Theorem states that u attains every complex value, with possibly one exception, in every...

On Finite Closures of Homogenized Solutions of Nonlinear Hyperbolic Equations

We study nonlinear equations subject to oscillatory initial data. The oscillatory solution of such problems tends to a homogenized weak limit that is characterized by the corresponding homogenized equations. Those equations usually involve an additional independent variable, so that the weak limit is an average of infinitely many functions. In certain cases, however, there is an alternative des...

Numerical Verifications of Solutions for Nonlinear Hyperbolic Equations

In this paper, we consider a numerical technique to enclose the solutions with guaranteed error bounds for nonlinear hyperbolic initial boundary value problems as well as to verify the existence of solutions. Using a finite element approximation and explicit error estimates for a certain simple linear hyperbolic problem, we construct, by computer, a set of functions which satisfies the conditio...