18 problems

Consider equation … with sufficiently large energy should blow up in finite time, in the sense that $w_{rr}(t,0)$ diverges as $t\nearrow T$ for some $T>0$. More precisely, there sh…

Consider equation … corresponding to sufficiently large initial data should blow up in finite time, in the sense that $w_{rr}(t,0)$ diverges as $t\nearrow T$ for some $T>0$, and th…

Scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_0=+\infty$, while in the focusing case ($\mu=-1$) we have $m_0=M(Q)$. This predicts global well-posedness and sc…

Delayed-blowup conjecture. The corresponding solution of either managed model does not blow up in $[0,T]$; thus, blowup can be delayed by dispersion or nonlinearity management.

Stable-manifold conjecture. Such initial data are globally well-posed under the nonlinear Schrödinger dynamics if and only if $u_0\notin\mathcal{W}^s(0)$.

Integral criterion conjecture. Small-data global existence holds if

Chen–Reissig's conjecture. If $C_{\mathrm{Str}}>0$, then the solution blows up at finite time for some initial data; if $C_{\mathrm{Str}}=0$, then small-data global existence holds…

Critical-threshold conjecture. For $d\in\{3,4,5,6\}$, the threshold of blowup is given by the codimension-one stable manifold of the self-similar solution $f_1$.

Strauss's conjecture. If $p>p_1(n)$, then small-data solutions exist globally; if $1<p<p_1(n)$, then small-data solutions blow up in finite time.

Let $\psi$ be a peak-type singular solution of the supercritical biharmonic nonlinear Schrödinger equation. Let $S_{\mathrm B}(\rho)$ be a radial self-similar profile satisfying ……

Equal-rate blowup conjecture. The solution undergoes equal-rate collapse,

Shrinking-ring conjecture. The solution is quasi self-similar, $\psi\sim\psi_{Q_{\mathrm B}}$, for $r-r_{\rm \max}=\mathcal{O}(L)$; it is a shrinking ring, with…

Let $\psi$ be a peak-type singular solution of the supercritical BNLS, let $L(t)$ be its collapsing length scale, let $T_{\rm c}$ be the collapse time, and set $\rho=r/L(t)$. Defin…

Consider the radially symmetric biharmonic nonlinear heat equation … and let $L(t)$ and $r_{\max}(t)$ denote the ring similarity scale and radius. Let $v_B(t,x)$ be a singular peak…

Let $\psi$ be a singular standing-ring solution of the BNLS with $d>1$ and $\sigma>4$. Let $L(t)$ be the ring-region similarity scale, $r_{\max}(t)$ its radius, and $\psi_B$ the ring p…

Let $\psi(t,r,\theta)=A(t,r)e^{imtheta}$ be a singular standing-vortex solution of the two-dimensional NLS with $\sigma>2$, and let $\psi_F$ denote the radial self-similar profile. Sup…

Let $\psi$ be a singular standing-ring solution of the NLS equation with $d>1$ and $\sigma>2$. Let $L(t)$ be its ring-region similarity scale, $r_{\max}(t)$ its ring radius, and…

Let $\dot{\mathcal H}^1$ be the wave-map energy space, and let an energy-class solution be a restriction of a maximal Cauchy development to a subinterval. Such a solution…