Related papers: Co-dimension one stable blowup for the supercritic…
We study singularity formation for the focusing quadratic wave equation in the energy supercritical case, i.e., for $d \geq 7$. We find in closed form a new, non-trivial, radial, self-similar blowup solution $u^*$ which exists for all $d…
We study the stability of an explicitly known, non-trivial self-similar blowup solution of the quadratic wave equation in the lowest energy supercritical dimension $d = 7$. This solution blows up at a single point and extends naturally away…
We consider the focusing cubic wave equation in the energy supercritical case, i.e., in dimensions $d \geq 5$. For this model an explicit nontrivial self-similar blowup solution was recently found by the first and third author in…
We consider the focusing nonlinear Schr\"odinger equation in three spatial dimensions with powers close to three and prove the existence of a self-similar solution. This generalizes a previous result on the cubic case and shows that…
We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in…
The focusing cubic wave equation in three spatial dimensions has the explicit solution $\sqrt{2}/t$. We study the stability of the blowup described by this solution as $t \to 0$ without symmetry restrictions on the data. Via the conformal…
We consider spherically symmetric supercritical focusing wave equations outside a ball. Using mixed analytical and numerical methods, we show that the threshold for blowup is given by a codimension-one stable manifold of the unique static…
We study finite-time blow-up for the one-dimensional nonlinear wave equation with a quadratic time-derivative nonlinearity, \[ u_{tt}-u_{xx}=(u_t)^2,\qquad (x,t)\in\mathbb R\times[0,T). \] Building on the work of Ghoul, Liu, and Masmoudi…
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…
We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial…
We consider the parabolic-elliptic Keller-Segel system in spatial dimensions $d\geq3$, which corresponds to the mass supercritical case. Some solutions become singular in finite time, an important example being backward self-similar…
We consider equivariant wave maps from $\mathbb{R}^{d+1}$ to $\mathbb{S}^d$ in supercritical dimensions $3\leq d\leq 6$. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable…
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…
We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…
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…
We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $d\geq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on…
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…
We consider the self-dual Chern-Simons-Schr\"odinger equation (CSS) under equivariance symmetry. Among others, (CSS) has a static solution $Q$ and pseudoconformal symmetry. We study the conditional stability of pseudoconformal blow-up…
In this note, we investigate the stability of self-similar blow-up solutions for superconformal semilinear wave equations in all dimensions. A central aspect of our analysis is the spectral equivalence of the linearized operators under…
We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution and we prove the asymptotic nonlinear stability of this solution in the whole space under…