Related papers: Fujita exponents on quantum Euclidean spaces
This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $\Omega$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$,…
Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for…
We consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction…
We consider the Cauchy problem of the nonlinear heat equation $u_t -\Delta u= u^{b},\ u(0,x)=u_0$, with $b\geq 2$ and $b\in \mathbb{N}$. We prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{n})$ (the Schwartz class)arbitrarily small…
The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a…
We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the "elastic" operator. We address local and global existence of solutions in two different regimes depending on the exponent in…
Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified…
We study linear pseudoparabolic equations with unbounded and time-dependent coefficients. We solve the case which has remained open in several recent studies of pseudoparabolic equations with unbounded and time-dependent coefficients. In…
We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-H\'enon equation $u_t - \Delta_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^{\gamma} u^p \mbox{ in } \mathbb{H}^N \times…
Recent progress in out-of-equilibrium closed quantum systems has significantly advanced the understanding of mechanisms behind their evolution towards thermalization. Notably, the concept of non-thermal fixed points (NTFPs) - responsible…
A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The…
We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first…
We prove existence and uniqueness of self-similar solutions with exponential form $$ u(x,t)=e^{\alpha t}f(|x|e^{-\beta t}), \qquad \alpha, \ \beta>0 $$ to the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta…
This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial…
We formulate non-relativistic classical and quantum mechanics in the non-commutative two dimensional plane. The approach we use is based on the Galilei group, where the non-commutativity is seen as a central extension upon identification of…
We present a modified cosmological scenario that arises from the application of non-extensive thermodynamics with varying exponent. We extract the modified Friedmann equations, which contain new terms quantified by the non-extensive…
In this paper, we study the following class of weighted Choquard equations \begin{align*} -\Delta u =\lambda u + \Bigg(\displaystyle\int\limits_\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ \Omega~~ \text{and}~~…
We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In…
In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness…
We consider general difference equations $u_{n+1} = F(u)_n$ for $n \in \mathbb{Z}$ on exponentially weighted $\ell_2$ spaces of two-sided Hilbert space valued sequences $u$ and discuss initial value problems. As an application of the…