659 research outputs found
On the Cauchy problem for a weakly coupled system of semi-linear -evolution equations with double dissipation
In this paper, we would like to consider the Cauchy problem for a
multi-component weakly coupled system of semi-linear -evolution
equations with double dissipation for any . The first main purpose
is to obtain the global (in time) existence of small data solutions in the
supercritical condition by assuming additional regularity for the initial
data and using multi-loss of decay wisely. For the second main one, we are
interested in establishing the blow-up results together with sharp estimates
for lifespan of solutions in the subcritical case. The proof is based on a
contradiction argument with the help of modified test functions to derive the
upper bound estimates. Finally, we succeed in catching the lower bound estimate
by constructing Sobolev spaces with the time-dependent weighted functions of
polynomial type in their corresponding norms.Comment: 19 page
L1-L1 estimates for a doubly dissipative semilinear wave equation
In this paper, we derive energy estimates and L1- L1 estimates, for the solution to the Cauchy problem for the doubly dissipative wave equation (Formula Presented.).The solution is influenced both by the diffusion phenomenon created by the damping term ut, and by the smoothing effect brought by the damping term - Δ ut. Thanks to these two effects, we are able to obtain linear estimates which may be effectively applied to find global solutions in any space dimension n≥ 1 , to the problems with power nonlinearities | u| p, | ut| p and | ∇ u| p, in the supercritical cases, by only assuming small data in the energy space, and with L1 regularity. We also derive optimal energy estimates and L1- L1 estimates, for the solution to the semilinear problems
On asymptotic properties of solutions to -evolution equations with general double damping
In this paper, we would like to consider the Cauchy problem for semi-linear
-evolution equations with double structural damping for any . The main purpose of the present work is to not only study the asymptotic
profiles of solutions to the corresponding linear equations but also describe
large-time behaviors of globally obtained solutions to the semi-linear
equations. We want to emphasize that the new contribution is to find out the
sharp interplay of ``parabolic like models" corresponding to and ``-evolution like models" corresponding to , which together appear in an equation. In this
connection, we understand clearly how each damping term influences the
asymptotic properties of solutions.Comment: 29 page
Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups
Let be a compact Lie group. In this article, we investigate the Cauchy
problem for a nonlinear wave equation with the viscoelastic damping on .
More preciously, we investigate some -estimates for the solution to the
homogeneous nonlinear viscoelastic damped wave equation on utilizing the
group Fourier transform on . We also prove that there is no improvement of
any decay rate for the norm by further assuming the
-regularity of initial data. Finally, using the noncommutative Fourier
analysis on compact Lie groups, we prove a local in time existence result in
the energy space Comment: 16 pages. arXiv admin note: text overlap with arXiv:2207.0442
Applications of estimates for solutions to semi-linear -evolution equations with general double damping
In this paper, we would like to study the linear Cauchy problems for
semi-linear -evolution models with mixing a parabolic like damping term
corresponding to and a -evolution like
damping corresponding to . The main goals are
on the one hand to conclude some estimates for solutions and their derivatives
in setting, with any , by developing the theory of
modified Bessel functions effectively to control oscillating integrals
appearing the solution representation formula in a competition between these
two kinds of damping. On the other hand, we are going to prove the global (in
time) existence of small data Sobolev solutions in the treatment of the
corresponding semi-linear equations by applying and
estimates, with and , from the
linear models. Finally, some further generalizations will be discussed in the
end of this paper.Comment: 38 page
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