Finite time blow-up for a class of parabolic or pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:

$u_t?a\Delta u_t?\Delta u+bu=k\left(t\right)|u{|}^{p?2}u,(x,t)\in \Omega \times (0,T),$

where $a\ge 0$, $b>?{?}_1$ with ${?}_1$ being the principal eigenvalue for $?\Delta$ on $H_0^1\left(\Omega \right)$ and $k\left(t\right)>0$. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) $J(u_0;0)<0$; (ii) $J(u_0;0)\le d\left(\infty \right)$, where $d\left(\infty \right)$ is a nonnegative constant; (iii) $0<J(u_0;0)\le C\rho \left(0\right)$, where $\rho \left(0\right)$ involves the $L^2$-norm or $H_0^1$-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.     

Nonexistence of global solutions for a class of nonlocal in time and space nonlinear evolution equations

In this paper, we study the nonlocal nonlinear evolution equation

${}^C{D}_{0|t}^{\alpha }u(t,x)?(J1\left|u\right|?\left|u\right|)(t,x){+}^C{D}_{0|t}^{\beta }u(t,x)={\left|u(t,x)\right|}^p,t>0,x\in {\mathbb{R}}^d,$

where $1<\alpha <2$, $0<\beta <1$, $p>1$, $J:{\mathbb{R}}^d\to {\mathbb{R}}_{+}$, $1$ is the convolution product in ${\mathbb{R}}^d$, and ${}^C{D}_{0|t}^q$, $q\in \{\alpha ,\beta \}$, is the Caputo left-sided fractional derivative of order $q$ with respect to the time $t$. We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when $1<p<1+\frac{2\beta }{d\beta +2(1?\beta )}$. Next, we deal with the system

where $1<\alpha <2$, $0<\beta <1$, $p>1$, and $q>1$. We prove that the system admitsnon global weak solution other than the trivial one with suitable initial data when $1<pq<1+\frac{2\beta }{d\beta +2(1?\beta )}max\{p+1,q+1\}$. Our approach is based on the test function method.     

Variational formulation of time-fractional parabolic equations

We consider initial/boundary value problems for parabolic PDE ${?}_t^{\alpha }u?\Delta u=f$ with fractional Caputo derivative ${?}_t^{\alpha }$ of order $1∕2<\alpha <1$ as time derivative and the usual Laplacian $?\Delta$ as space derivative (also called fractional diffusion equations in the literature). We prove well-posedness of corresponding variational formulations based entirely on fractional Sobolev–Bochner spaces, and extend existing results for possible choices of the initial value for $u$ at $t=0$.