Consider the semilinear heat equation ∂tu = ∂ xu + λσ(u)ξ on the interval [0 , 1] with Dirichlet zero boundary condition and a nice nonrandom initial function, where the forcing ξ is space-time white noise and λ > 0 denotes the level of the noise. We show that, when the solution is intermittent [that is, when infz |σ(z)/z| > 0], the expected L-energy of the solution grows at least as exp{cλ} an...

In the present paper, we investigate the preventive role of space dimension for semilinear parabolic problems. Conditions guaranteeing the absence of the blow-up of the solutions are formulated.

For a class of semilinear parabolic equations, we prove both global existence and finite-time blow-up depending on the initial datum. The proofs involve tools from the potential-well theory, from the criticalpoint theory, and from classical comparison principles.

Abstract The boundary value problem for semilinear parabolic stochastic equations of the form dX − ∆X dt + β(X)dt 3 QdWt, where Wt is a Wiener process and β is a maximal monotone graph everywhere defined, is well posed.

We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation ut = ∆u + up in RN , where p > 1 + 2/N , p(N − 2) ≤ N + 2. The initial data are of the form u(x, 0) = αφ(x), where φ is a fixed function with suitable decay at |x| = ∞ and α > 0 is a parameter. There exists a threshold parameter α∗ such that the solution exists globally if and o...

We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations ut = ∆u + f(u) on RN . Here f ∈ C1, f(0) = 0, and a localized solution refers to a solution u(x, t) which decays to 0 as x → ∞ uniformly with respect to t > 0. In all previously known examples, bounded, localized solutions are convergent or at least quasiconvergent in the sense that all their limi...

We deal with a class of semilinear parabolic PDEs on the space continuous functions that arise, for example, as Kolmogorov equations associated to infinite-dimensional lifting path-dependent SDEs. investigate existence smooth solutions through their representation via forward–backward stochastic systems, which we provide necessary regularity theory. Because lack smoothing properties operators a...

The semilinear normal parabolic equations corresponding to 3D Navier-Stokes system have been derived. The explicit formula for solution of normal parabolic equations with periodic boundary conditions has been obtained. It was shown that phase space of corresponding dynamical system consists of the set of stability (where solutions tends to zero as time t → ∞), the set of explosions (where solut...

Under suitable controllability and smoothness assumptions, the Minimum Time function T (x) of a semilinear control system is proved to be locally Lipschitz continuous and semicon-cave on the controllable set. These properties are then applied to derive optimality conditions relating optimal trajectories to the superdiierential of T .