| In this thesis, we mainly consider the existence, uniqueness and the long time behavior of the solutions to the following two reaction diffusion equations with non-regular data. where u0,g∈L1(Ω) or a Radon measure independent of time.For the first problem, we provide the existence of the solution to the problem and its corresponding elliptic problem. We prove also the uniqueness of the so-lution and the continuity of the solutions semigroup. Using the decomposition technique combined with the bootstrap method, we establish some new regular-ity results of the solution. The arguments we used here help us overcome, suc-cessfully, the difficulties brought by the nonregular forcing term and initial data Concerning the long time behavior of the solution, we obtain the existence of the global attractor (?) in L1(Ω). Then we prove the semigroup of the trans-lated problem possesses a global attractor (?)v in Lr(Ω)∩H10(Ω),1≤r<∞. At last, we obtain that the attractor (?) is actually invariant and compact in < max{N/(N-1), (2p-2)/p}, and attracts every bounded sets in L1(Ω) in the topology of Lr(Ω)∩H01(Ω),1≤r<∞. If g is a Radon measure absolutely continuous with the parabolic capacity and independent of time, we get similar results as above. All the results hold for f(x, u) satisfying similar conditions.For the second problem, we first give the existence results of the prob-lem itself and its corresponding elliptic equation. Then we give short proof of the uniqueness of the solution to the parabolic problem. We point that the existence results have been obtained in the literatures in the case f is monotone. Yet, nobody considered the uniqueness of the solution in our case. With the existence results, we use again the decomposition and boot-strap technique to establish the regularity results of the solution. For the long time behavior of the solution, similarly, we prove that the solution semigroup possesses a global attractor (?) in L1(Ω), which is invariant and compact in Lr-1(Ω)∩W01,s(Ω),s |
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