Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A New Condition for the Concavity Method of Blow-up Solutions to
  Semilinear Heat Equations   In this paper, we consider the semilinear heat equations under Dirichlet
boundary condition \[ u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)),
& \left(x,t\right)\in \Omega\times\left(0,+\infty\right), u\left(x,t\right)=0,
& \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right),
u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \] where $\Omega$ is a
bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary
$\partial\Omega$. The main contribution of our work is to introduce a new
condition \[ (C) \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta
u^{2}+\gamma,\,\,u>0 \] for some $\alpha, \beta, \gamma>0$ with
$0<\beta\leq\frac{\left(\alpha-2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is
the first eigenvalue of Laplacian $\Delta$, and we use the concavity method to
obtain the blow-up solutions to the semilinear heat equations. In fact, it will
be seen that the condition (C) improves the conditions known so far.

37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas