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Hypercontractivity, HopfLax type formulas, OrnsteinUhlenbeck operators (II)
Global existence and exponential decay rates for the Westervelt equation
1.  Department of Mathematics, University of Stuttgart, 70569 Stuttgart, Germany 
2.  University of Virginia, Department of Mathematics, Charlottesville, VA 22901 
Our main results are : (1) global wellposedness, (2) exponential decay rates for the energy function corresponding to both weak and strong solutions. The proof is based on (i) application of a suitable fixed point theorem applied to an appropriate formulation of the PDE which exhibits analyticity properties of the underlying linearised semigroup, (ii) exploitation of decay rates associated with the dissipative mechanism along with barrier's method leading to global wellposedness. The obtained result holds for all times, provided that the initial data are taken from a suitably small ball characterized by the parameters of the equation.
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