## Rocky Mountain Journal of Mathematics

### The heat equation for local Dirichlet forms: Existence and blow up of nonnegative solutions

Tarek Kenzizi

#### Abstract

We establish conditions ensuring either existence or blow up of nonnegative solutions for the following parabolic problem: $$\begin{cases} Hu -Vu + ({\partial u}/{\partial t}) =0 & \mbox {in } X\times (0,T), \\ u(x,0)=u_{0}(x) & \mbox {in } X, \end{cases}$$ where $T>0$, $X$ is a locally compact separable metric space, $H$ is a selfadjoint operator associated with a regular Dirichlet form $\mathcal E$; the initial value $u_{0}\in L^{2}(X,m)$, where $m$ is a positive Radon measure on Borel subset $U$ of $X$ such that $m(U)>0$ and $V$ is a Borel locally integrable function on $X$.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2573-2593.

Dates
First available in Project Euclid: 30 December 2018

https://projecteuclid.org/euclid.rmjm/1546138822

Digital Object Identifier
doi:10.1216/RMJ-2018-48-8-2573

Mathematical Reviews number (MathSciNet)
MR3894994

Zentralblatt MATH identifier
1403.35120

#### Citation

Kenzizi, Tarek. The heat equation for local Dirichlet forms: Existence and blow up of nonnegative solutions. Rocky Mountain J. Math. 48 (2018), no. 8, 2573--2593. doi:10.1216/RMJ-2018-48-8-2573. https://projecteuclid.org/euclid.rmjm/1546138822

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