Related papers: Determine the spacial term of a two-dimensional he…
We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.
This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic…
In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients $f$ and $g^k$, driven by a sequence $(\beta^k)_k$ of i.i.d.…
We solve an inverse problem for the one-dimensional heat diffusion equation. We reconstruct the heat source function for the three types of data: 1) single position point and different times, 2) constant time and uniformly distributed…
In this paper, we study the formation of finite time singularities for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first spacial derivative of the solution…
We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $\sigma$-finite measure…
For the fractional heat equation $\frac{\partial}{\partial t} u(t,x) = -(-\Delta)^{\frac{\alpha}{2}}u(t,x)+ u(t,x)\dot W(t,x)$ where the covariance function of the Gaussian noise $\dot W$ is defined by the heat kernel, we establish…
This note is devoted to a study of $L^q$-tracing of the fractional temperature field $u(t,x)$ -- the weak solution of the fractional heat equation $(\partial_t+(-\Delta_x)^\alpha)u(t,x)=g(t,x)$ in $L^p(\mathbb R^{1+n}_+)$ subject to the…
We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we…
In this paper, we investigate a nonlinear inverse problem aimed at recovering a coefficient $a(t, x)$, dependent on both time and a subset of spatial variables, in a diffusion equation \( u_t - \Delta_x u - u_{yy} +a(t, x) u = f(t,x,y) \),…
In this paper, we represent the exact solution of a two phase inverse spherical Stefan problem, where along with unknown temperature functions heat flux function has to be determined. Suggested solution is obtained from new form of integral…
We give an explicit representation of the fundamental solution to the heat equation on a half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition, and obtain upper and lower estimates of the fundamental solution.…
We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- \Delta)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$,…
We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the…
In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and…
This work investigates the two-dimensional thermal behavior of a bilayer medium subject to both internal and external heat sources. The model incorporates diffusion, advection, and temperature-dependent volumetric heat generation or…
We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension three $$ u_t = \Delta u + u^5 , \quad {\mbox {in}} \quad \R^3 \times (0,\infty), \ \ u(x, 0)= u_0 (x)\inn \R^3. $$ For…
In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = \Delta u +a |u|^\alpha u, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; \alpha>0;$ with initial value $u(0)\in…
Four problems about recovery of a high-frequency source in the one-dimension heat equation with homogeneous initial-boundary conditions by some information about partial asymptotic of its solution have solved. It is shown, that the source…
We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$…