On solvability of a class of degenerate Kirchhoff equations with logarithmic nonlinearity
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We study the Dirichlet problem for the degenerate nonlocal parabolic equation \[ u_{t}-a\left(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2}\right)\Delta u=C_b\left\Vert u\right\Vert _{L^2(\Omega)}^{\beta}\left\vert u \right\vert^{q\left(x,t\right)-2}u\log|u|+f \quad \text{in $Q_T$}, \] where $Q_{T}:=\Omega \times (0,T)$, $T>0$, $\Omega \subset \mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with a sufficiently smooth boundary, $q(x,t)$ is a measurable function in $Q_{T}$ with values in an interval $[q^{-},q^{+}]\subset(1,\infty)$ and the diffusion coefficient $a(\cdot)$ is a continuous function defined on $\mathbb{R}_+$. It is assumed that $a(s)\to 0$ or $a(s)\to \infty$ as $s\to 0^+$, therefore the equation degenerates or becomes singular as $\|\nabla u(t)\|_{2}\to 0$. For both cases, we show that under appropriate conditions on $a$, $\beta$, $q$, $f$ the problem has a global in time strong solution which possesses the following global regularity property: $\Delta u\in L^2(Q_T)$ and $a(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2})\Delta u\in L^2(Q_T)$.
Abstract
We study the Dirichlet problem for the degenerate nonlocal parabolic equation \[ u_{t}-a\left(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2}\right)\Delta u=C_b\left\Vert u\right\Vert _{L^2(\Omega)}^{\beta}\left\vert u \right\vert^{q\left(x,t\right)-2}u\log|u|+f \quad \text{in $Q_T$}, \] where $Q_{T}:=\Omega \times (0,T)$, $T>0$, $\Omega \subset \mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with a sufficiently smooth boundary, $q(x,t)$ is a measurable function in $Q_{T}$ with values in an interval $[q^{-},q^{+}]\subset(1,\infty)$ and the diffusion coefficient $a(\cdot)$ is a continuous function defined on $\mathbb{R}_+$. It is assumed that $a(s)\to 0$ or $a(s)\to \infty$ as $s\to 0^+$, therefore the equation degenerates or becomes singular as $\|\nabla u(t)\|_{2}\to 0$. For both cases, we show that under appropriate conditions on $a$, $\beta$, $q$, $f$ the problem has a global in time strong solution which possesses the following global regularity property: $\Delta u\in L^2(Q_T)$ and $a(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2})\Delta u\in L^2(Q_T)$.
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