This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity u log |u| -ffl u log |u| dx in a bounded domain, subject to homogeneous Neumann boundary value condition. By using the logarithmic Sobolev inequality and energy estimate methods, we get the results under appropriate conditions on blow-up and non-extinction of the solutions, which extend some recent results.

In this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

In this paper, we deal with the following quasilinear attraction-repulsion model: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u t = ∇ • (D(u)∇u)-∇ • (S(u)χ (v)∇v) + ∇ • (F(u)ξ (w)∇w) + f (u), x ∈ Ω, t > 0, v t = v + βuαv, x ∈ Ω, t > 0, 0 = w + γ u-δw, x ∈ Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω with homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 2). Let the chemotactic sensitivity χ (v) be a positive constant, and let the chemotactic sensitivity ξ (w) be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.

We consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic-elliptic type with logistic source under homegeneous Neumann boundary conditions in a bounded domain `\Omega\subset R^{n}(n\geq2)` with smooth boundary, where`D(u)\geq c_{D}(u+1)^{m-1}` with `m\geq1`and `c_{D}>0`, `f(u)\leq a-bu^{\eta}` with `\eta>1`.{ We show two cases that the system admits a uniqueglobal bounded classical solution depending on `0\leq S(u)\leq C_{s}(u+1)^{q}, 0\leq F(u)\leq C_{F}(u+1)^{g}` by Gagliardo-Nirenberg inequality.For specific `D(u),S(u),F(u)` with logistic source for `\eta>1` and `n=2`, we establish the finite time blow-up conditions forsolutions that the finite time blow-up occurs at `x_{0}\in\Omega` whenever `\int_{\Omega}u_{0}(x)dx>\frac{8\pi}{\chi\alpha-\xi\gamma}`with `\chi\alpha-\xi\gamma>0`, under `\int_{\Omega}u_{0}(x)|x-x_{0}|^{2}dx` sufficiently small.