12 problems

Let $\Omega$ be a domain and let $u$ solve the semilinear Dirichlet problem … Here, $u$ is quasiconcave when its superlevel sets are convex. Lions' conjecture. For a general $f$, t…

Let $f$ be a differentiable reaction term whose derivative is discontinuous, and consider the finite element approximation experiment described in the source, in which the iterativ…

The paper considers the more general problems referred to as … , in the one-dimensional case $N=1$. Conjecture. A similar strategy to the one used for the ball case with $N=1$ and…

Let $u$ solve the one-dimensional initial value problem … with compactly supported smooth initial data $u(x,0)=\varepsilon f(x)$ and $u_t(x,0)=\varepsilon g(x)$, where $p,q,r>1$,…

Lions's conjecture. For a general $f$, the superlevel sets of $u$ should be convex.

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$, and let $u$ solve … Assume that $u$ has finitely many critical points and that $u(x)>0$ implies $x_1>0$. Half-space rigidity conj…

Takamura's conjecture. The lifespan satisfies

The paper considers the fractional elliptic problem … with $s\in(\frac{1}{2},1)$, nondecreasing $f\in C(\mathbb R)$ satisfying $t f(t)\geq 0$, and data $\tau\in L^1(\Omega,\rho^s)$…

Consider a bounded convex plane domain $\Omega$ and a positive solution $u$ of … where the nonlinearity $f$ is Lipschitz and restoring, meaning that $f(z)>0$ whenever $z>0$. Cima–D…

Let $L_{\gamma}=\frac{2\pi}{\sqrt{\lambda_{\gamma}}}$ be the period arising from the $k=0$ projection of the linearization around the constant solution $v_1\equiv 1$, where…

For the scalar semilinear equation, let $n$ denote the spatial dimension, $m$ the physical mass, and let the nonlinearity exponent satisfy $\alpha\in(0,\infty)$. Forbidden physical…

Generic blow-up conjecture. Based on numerical evidence, this explicit solution, after smooth cut-off and finite-speed propagation, describes the generic blow-up scenario for solut…