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Global existence and exponential decay rates for the Westervelt equation
We consider the Westervelt equation which models propagation of sound in a fluid medium.
This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging and therapy.
The PDE model consists of the second order in time evolution
which is both quasilinear and degenerate. Degeneracy depends on the
fluctuations of the acoustic pressure.
Our main results are : (1) global well-posedness,
(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.