Contribution to volume
D. Amadori, E. Dal Santo, F. Aqel
Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain
J. Math. Pures Appl. 132 (2019), 166-206
In this paper we study the long time behavior for a semilinear wave equation
with space-dependent and nonlinear damping term.
After rewriting the equation as a first order system, we define a class of approximate solutions
that employ tipical tools of hyperbolic systems of conservation laws, such as the Riemann problem.
By recasting the problem as a discrete-time nonhomogeneous system, which is related to a probabilistic interpretation of the solution,
we provide a strategy to study its long-time behavior uniformly with
respect to the mesh size parameter 1/N -> 0.
The proof makes use of the Birkhoff decomposition of doubly stochastic
matrices and of accurate estimates on the iteration system as N-> ∞.
Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in L∞ of the solution to the first order system
towards a stationary solution, as t->∞, as well as uniform error estimates for the approximate solutions.
D. Amadori, R.M. Colombo, G. Guerra, W. Shen
Slow Erosion of Granular Flow: Continuous and Discontinuous Profiles
Proc. HYP2012, pp.
Global BV solutions and relaxation limit for a system of conservation
Proc. Roy. Soc. Edinburgh Sect. A 131
(On) Global weak solutions for systems of balance laws
Appl. Math. Lett. 12 (1999), 123-127
P. Baiti, P.G. LeFloch, B. Piccoli
Non-Classical Shocks and the Cauchy Problem for Nonconvex Conservation
J. Differential Equations 151 (1999), 345-372
Viscosity Solutions and Standard Riemann Semigroup for Conservation
Laws with Boundary
Rend. Sem. Mat. Univ. Padova 99 (1998), 219-245
Continuous Dependence for 2x2 Conservation Laws with Boundary
J. Differential Equations 138 (1997),
problems for systems of conservation laws
NoDEA 4 (1997), 1-42
Unstable Blow-up Patterns
Diff. and Integral Equations, 8 (1995), 1977-1996
A Class of Hyperbolic Operators with Double Characteristics
Comm. Partial Differential Equations 19 (1994), 1185-1201