Stochastic heat equations with logarithmic nonlinearity

A local absorbing boundary condition for 3D seepage and heat transfer in unbounded domains

“…To prove the solution of the problem above unique and stable, it is needed to prove that the unknown variable satisfies the following Grönwall's Lemma. 24 ‖𝑢‖…”

“…where, 𝑔(𝑡) = ‖𝑢 0 ‖ 2 Ω + ‖𝑓‖ 2 Ω×T . According to the Grönwall's Lemma, 24 Equation (40) satisfies the Equation (30). Thus, the reduced initial-boundaryvalue problem is stable.…”

“…To prove the solution of the problem above unique and stable, it is needed to prove that the unknown variable satisfies the following Grönwall's Lemma 24 $$$$\begin{equation}\left\| u \right\|_{\Omega \times {\mathrm{T}}}^2 \le {e}^t\left( {\left\| {{u}_0} \right\|_\Omega ^2 + \int_{{\mathrm{T}}}{{g\left( \tau \right){\mathrm{d}}\tau }}} \right)\end{equation}$$$$where, T denotes the time scope [0, t ]; Ω denotes the space scope x × y × z = [0, +∞) × (−∞, +∞)×(−∞, +∞);$${\| \cdot \|}_\Omega = {( {\int_{\Omega }{{{{( \cdot )}}^2{\mathrm{d}}x\,{\mathrm{d}}y\,{\mathrm{d}}z}}} )}^{\frac{1}{2}}$$;$${\| \cdot \|}_{\Omega \times {\mathrm{T}}} = {( {\int_{{\mathrm{T}}}{{\int_{\Omega }{{{{( \cdot )}}^2{\mathrm{d}}x\,{\mathrm{d}}y\,{\mathrm{d}}z}}\,{\mathrm{d}}\tau }}} )}^{\frac{1}{2}}$$;$$g( t ) = \| {{u}_0} \|_\Omega ^2 + \int_{{\mathrm{T}}}{{\int_{\Omega }{{{f}^2{\mathrm{d}}x{\mathrm{d}}y{\mathrm{d}}z}}{\mathrm{d}}\tau }}$$.…”

Global solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

“…Mueller's results have been generalized to other settings including fractional heat equations [2,10], nonlinear Schrödinger equation [8], and stochastic wave equation [19]. More recently, researchers have investigated the effects that adding superlinear deterministic forcing terms f (u(t, x)) to the right-hand side of (1.1) has on the finite time explosion of the stochastic heat equation [1,4,7,9,11,13,15,24,25]. Similar explosion problems have been investigated for the stochastic wave equation [12,16].…”