I consider the nonlinear heat equation $${u_t} - \Delta u = {\left| u \right|^{p - 2}}u,\,\left( {x,t} \right) \in \left( {0,1} \right) \times \left( {0,T} \right)$$ with Dirichlet boundary conditions and initial condition $u\left( {x,0} \right) = {u_0}\left( x \right)$. My question is how can we extension the solution of this equation with some appreciate conditions? Because the nonlinear term in RHS so i cant estimate the solution is bounded. I don't have any idea to solve it.


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