This paper delves into subwavelength resonances within three dimensional elastic scattering systems. These systems are made up of multi-layered, high-contrast nested resonator structures with arbitrary shapes. Initially, we use the Dirichlet-to-Neumann map and a variational method to describe these resonances. We show that they arise when the determinant of a particular matrix becomes zero. Secondly, by applying the Gohberg–Sigal theory and Puiseux expansions, we characterize the subwavelength characteristic values near the origin and derive their asymptotic expansions. For an N-layer structure, the number of subwavelength characteristic values is 12N, counted with algebraic multiplicity, including 6N branches with positive real parts and 6N branches with negative real parts. Finally, we offer a modal decomposition formula for the total field of the elastic scattering problem and obtain the transverse and longitudinal far-field patterns related to the elastic far-field modes scattered by the N-layer nested resonators.
In this paper, we investigate some nonlocal Whitham-type equations which include the nonlinear dispersive θ-equation, the Fornberg-Whitham-Degasperis-Procesi equations and the generalized rod equation. A class of priori estimates of L∞-norm of the solution to the corresponding equations is firstly constructed. Then, a novel global existence and two wave-breaking criteria involving different range of the parameter θ to the nonlinear dispersive θ-equation are obtained without relying on any conservation law properties. Subsequently, comparing with the results of previous references Shu (2025) [54], we consider some new wave-breaking criteria of the Fornberg-Whitham-Degasperis-Procesi equations depending on the piecewise intervals of the parameter ba. Finally, we obtain the global existence for the generalized rod equation with the parameters βγ0∈[2,3]. Some wave-breaking criteria of the generalized rod equation are studied by discussing the specific range of βγ0, which refines and extends the results from previous references Zhou and Lai (2024) [64].
In this paper, we study the following first-order Hamiltonian system with an L2-constraint:{−Jz˙−Az=|K(t)z|p−2K(t)TK(t)z+λzinR,∫R|z|2dt=c>0, where z=(z1,...,z2N)∈R2N, λ∈R appears as a Lagrange multiplier, and J, A, K(t) are 2N×2N real matrices with J−1=JT=−J, A=AT. Using variational methods for strongly indefinite problems, we prove the existence of ground state homoclinic orbits for all p∈(2,6). In particular, the number p¯:=4 is referred to as the mass-critical exponent. The case p∈[4,6), which covers the mass-critical (p=4) and part of the mass-supercritical (4<p<6) regimes, poses considerable difficulties for existing methods and constitutes the main focus of this paper. We also investigate some properties of these homoclinic orbits, including the compactness of the set of ground states and the exponential decay of ground state homoclinic orbits. Furthermore, bifurcation phenomena related to this problem are discussed, namely the asymptotic behavior of ground state homoclinic orbits and the corresponding Lagrange multipliers as c→0+. The results obtained in this paper to some extent supplement and extend the results presented in Guo and Yu (2024) [24].
In this paper, we develop an analytical method to prove the sharp upper and lower bounds for the first eigenvalue of the so-called Euler-Bernoulli beam equations, provided the L1-norms and the average of the potentials. In particular, we focus on the indefinite potentials. We also study a kind of inverse spectral problems, which means that the optimal lower bounds of L1-norms of the potentials can be obtained when the first eigenvalue and the average of the potentials are fixed. We use the setting of measure differential equations to understand the direct spectral problems as well as the inverse problems because the solutions for them will lead to Dirac measure distributions of potentials.
In this paper, we study the existence of traveling wave to a cross-diffusion and degenerate reaction-diffusion system. The underlying biological problem is to describe the process in microscopic scale the invasion of cancer cells to the damage of healthy tissues. The mathematical model is established by renowned biologist and cancer expert Gatenby and his collaborator Gawlinski [10], and later generalized in [19] to cover wide range of parameter setting and more intricate interactions between the underlying mathematical quantities which represent tumor density, acid concentration and healthy tissue density. Mathematically, the cancer invasion phenomenon is quantitatively depicted by the existence of a traveling wave solution connecting two equilibrium states, one of which is the state of normal tissue without cancer cells, and the other with presence of cancer cells, and normal tissue being either partially or completely eliminated (depending on the range of parameters). We prove the existence of traveling wave for all speed bigger or equal to some value, for a rather general setting of parameters to cover both cases of either normal tissue is partially or completely eliminated after cancer invasion.
