Related papers: Control of blow-up singularities for nonlinear wav…
We consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it. At noncharacteristic points on the singularity surface, Merle and Zaag have identified the rate of blow-up of the $H^1$-norm of…
This work investigates radial solutions for nonlinear fractional Schr\"odinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the…
In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective…
In this article, we investigate certain theoretical aspects of the hierarchical controllability problem in one-dimensional wave equations within a moving domain using Stackelberg strategy. The controls are applied along a portion of the…
We prove local exact controllability in arbitrary short time of the two-dimensional incompressible Euler equation with free surface, in the case with surface tension. This proves that one can generate arbitrary small amplitude periodic…
We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y -…
We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is…
Idealized networks of integrate-and-fire neurons with impulse-like interactions obey McKean-Vlasov diffusion equations in the mean-field limit. These equations are prone to blowups: for a strong enough interaction coupling, the mean-field…
The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a…
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the…
Bose-Einstein condensation is usually modeled by nonlinear Schroedinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For…
In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) =…
We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground…
We consider an elliptic equation in a cone, endowed with (possibly inhomogeneous) Neumann conditions. The operator and the forcing terms can also allow non-Lipschitz singularities at the vertex of the cone. In this setting, we provide…
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…
In this paper, we discuss the Cauchy Problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. Firstly, we establish the local existence of regular solutions by the fundamental methods in the…
In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness…
We consider the one-dimensional Euler-Poisson system equipped with the Boltzmann relation and provide the exact asymptotic behavior of the peaked solitary wave solutions near the peak. This enables us to study the cold ion limit of the…
We consider a class of blow-up solutions for perturbed nonlinear heat equations involving gradient terms. We first prove the single point blow-up property for this equation and determine its final blow-up profile. We also give a sharper…
We investigate the blow-up dynamics of smooth solutions to the one-dimensional wave equation with a quadratic spatial derivative nonlinearity, motivated by its applications in Effective Field Theory (EFT) in cosmology. Despite its…