Abstract
We study the boundedness of global solutions to the semilinear parabolic equations with combined nonlinearities. Firstly, we establish a sufficient condition for the initial data using the potential well method, ensuring the global existence of solutions. Subsequently, we proceed to demonstrate an a priori estimate for all global solutions. Finally, the potential well theory, in conjunction with the derived a priori estimate, we show that there is a sub-convergent sequence of the global flow, which converges the positive solution to the corresponding stationary equation.
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