27 research outputs found
Decay Rates of Solutions for Non-Degenerate Kirchhoff Type Dissipative Wave Equations
Consider the Cauchy problem for the non-degenerate Kirchhoff type dissipative wave equations with the initial data belonging to (H2(RN)ā©L1(RN))Ć(H1(RN)ā©L1(RN)). Using the Fourier transform method in the L2 ā© L1-frame, we can improve the decay rates of the energies given by the energy method of the L2-frame
Upper Decay Estimates for Non-Degenerate Kirchhoff Type Dissipative Wave Equations
We study on the Cauchy problem for non-degenerate Kirchhoff type dissipative wave equations Ļuā²ā² + a (||A1/2u(t)||2) Au + uā² = 0 and (u(0), uā²(0)) = (u0, u1), where u0 ā 0 and the nonlocal nonlinear term a(M) = 1+MĪ³ with Ī³ > 0. Under the suitably smallness condition, we derive the upper decay estimates of the solution u(t) for the case of 0 < Ī³ < 1 in addition to Ī³ ā„ 1
Kirchhoff Type Dissipative Wave Equations in Unbounded Domains
Consider the Cauchy problem for the non-degenerate Kirchhoff type dissipative wave equations with the initial data belonging to H2(RN)ĆH1(RN) in unbounded domains. When the coefficient Ļ or the initial energy E(0) is small at least, we show the global existence theorem and derive decay estimates of energies in the L2-frame. Moreover, when the initial data belong to L1(RN)ĆL1(RN) in addition, we improve the decay rates of the solutions
Lower Decay Estimates for Non-Degenerate Kirchhoff Type Dissipative Wave Equations
We consider the Cauchy problem for non-degenerate Kirchhoff type dissipative wave equations Ļuā²ā²+ a (ā„A1/2u(t)ā„2) Au + uā² = 0 and (u(0), uā²(0)) = (u0, u1), where u0 ā 0. We derive the lower decay estimate ā„u(t)ā„2 ā„ CeāĪ²t for t ā„ 0 with Ī² > 0 for the solution u(t)
Kirchhoff equations with strong damping
We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the āelasticā operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space, assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space, we prove local existence and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces
Nonlinear Partial Differential Equations on Graphs
One-dimensional metric graphs in two and three-dimensional spaces play an important role in emerging areas of modern science such as nano-technology, quantum physics, and biological networks. The workshop focused on the analysis of nonlinear partial differential equations on metric graphs, especially on the bifurcation and stability of nonlinear waves on complex graphs, on the justification of Kirchhoff boundary conditions, on spectral properties and the validity of amplitude equations for periodic graphs, and the existence of ground states for the NLS equation with and without potential
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs