In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2. To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term.
AbstractIn this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.
Abstract
We study an anisotropic, possibly non-homogeneous version of the evolution 𝑝-Laplacian equation when fast diffusion holds in all directions.
We develop the basic theory and prove symmetrization results from which we derive sharp
L
1
L^{1}
-
L
∞
L^{\infty}
estimates.
We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range.
We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution.
Positivity, decay rates as well as other properties of the solutions are derived.
The combination of self-similarity and anisotropy is not common in the related literature.
It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.
Let $\mathcal{S}$ be the Sierpi\’nski
gasket in $\mathbb{R}^2$ and
$\mathcal{S}_{0}$ denote the boundary of
$\mathcal{S}$. In this paper, we study the following
non-homogeneous $p$-Laplacian equation \begin{align*}
-\Delta_p u &= \lambda
|u|^{q-2} u + f
\text{~in}\;
\mathcal{S}\setminus\mathcal{S}_0\\
u &=
0\;\mbox{~on}\;
\mathcal{S}_0, \end{align*} where
$p$, $q$, $\lambda$ are real numbers such that
$\lambda >0$, $1
In this paper, we consider the following p-Laplacian equation
−∆pu + |u|p−2u − λu = µ|u|q−2u +
|u|p∗−2u, in RN, u > 0, ∫ RNu2dx = a2,
where a,µ > 0, −∆pu = div(|∇u|p−2∇u),1
< p < N, λ ∈ R is an unknown parameter that appears
as a Lagrange multiplier, p < q
Growing solutions of the fractional p-Laplacian equation in the Fast Diffusion range