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In this paper, we consider the wave equation with variable coefficients and boundary damping and supercritical source terms. The goal of this work is devoted to prove the local and global existence, and classify decay rate of energy depending on the growth near zero on the damping term. Moreover, we prove the blow-up of the weak solution with positive initial energy as well as nonpositive initial energy.

In this paper, we study the blow-up radial solution of fully parabolic system with higher dimensional two species Cauchy problem for some initial condition. In addition, we show that the set of positive radial functions in $L^{1}(\mathbb{R})\cap BUC(\mathbb{R}^{N}) \times L^{1}(\mathbb{R})\cap BUC(\mathbb{R}^{N}) \times W^{1,1}(\mathbb{R}^{N}) \cap W^{1,\infty}(\mathbb{R}^{N})$ has a dense subset composed of positive radial initial data causing blow-up in finite time with respect to topology $L^{p}(\mathbb{R}^{N}) \times L^{p}(\mathbb{R}^{N}) \times H^{1}(\mathbb{R}^{N})\cap W^{1,1}(\mathbb{R}^{N})$ for $p \in \left[1,\frac{2N}{N+2}\right)$.

In this paper, we consider the following attraction repulsion chemotaxis model with nonlinear signal term: \begin{align*} &u_{t}=\nabla \cdot(\nabla u-\xi_{1} u \nabla v +\xi_{2} u \nabla w), \quad &0=\Delta v -\lambda_{1}v +f_{1}(u), \quad &0=\Delta w -\lambda_{2}w +f_{2}(u), \quad x \in \mathbb{R}^{N}, t>0, \end{align*} where $\xi_{1},\xi_{2},\lambda_{1},\lambda_{2}$ are for some positive constants, and \begin{equation*} f_{1} \in C^{1}([0,\infty)) \; \text{satisfying} \; 0 \leqslant f_{1}(s) \leqslant c_{1}s^{l}, \; \forall s \geqslant 0 \ \text{and} \ l> 0, \end{equation*} \begin{equation*} f_{2} \in C^{1}([0,\infty))\; \text{satisfying} \; 0 \leqslant f_{2}(s) \leqslant c_{2}s^{m}, \; \forall s \geqslant 0 \ \text{and} \ m> 0. \end{equation*} We will show that this problem has a unique global bounded solution when $ l>\frac{2}{N}, l