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We study positive blowing-up solutions of systems of the form: $$u_t=delta_1 Delta u+e^{pv},quad v_t= delta_2Delta v+e^{qu},$$ with $delta_1,delta_2>0$ and $p, q>0$. We prove single-point blow-up for large classes of radially decreasing solutions. This answers a question left open in a paper of Friedman and Giga~(1987), where the result was obtained only for the equidiffusive case $delta_1=delta_2$ and the proof depended crucially on this assumption.
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclide
We study the existence of nontrivial solutions for a nonlinear fractional elliptic equation in presence of logarithmic and critical exponential nonlinearities. This problem extends [5] to fractional $N/s$-Laplacian equations with logarithmic nonlinea
The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-S
We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $rho(x)$ and a power-like reaction term. We show that for small enough initial data, if $rho(x)sim frac{1}{lef
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the gradient of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: