Regularity and time behavior of the solutions to weak monotone parabolic equations

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0$$\end{document} is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0$$\end{document}, immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

Nevertheless, we show in this paper some unknown surprising properties of the solutions. The first issue we want to address stems from the following unexpected and incredible result: even if the initial datum u 0 is only a summable function, if the reaction term f is the zero function and p is "not too small" (i.e., p > 2N N +r 0 ), then there exists a solution u of (1.1) which "immediately" becomes bounded, i.e., such that u(t) ∈ L ∞ ( ), for every t ∈ (0, T ). Moreover, it is also possible to estimate the L ∞ -norm of u(t) but these estimates differ with p and exhibit a different behavior in the degenerate case from the singular one. In particular, in the degenerate case p > 2 the following decay estimate holds true where C is a constant depending only on α, N , | | and p (see [14,42] and the references therein) and hence is an estimate that doesn't depend on the initial datum u 0 and for this reason it is also called universal (decay) estimate. We point out that the universal estimate (1.9) is a peculiarity of the degenerate case (in a bounded domain) and it is no longer true in the singular case p < 2 or for p = 2.
If p = 2, the same decay of the heat equation appears since the following exponential estimate holds true where σ is a positive constant depending only on α, N , r 0 and . Finally, if p > 2N N +r 0 , hence in both the previous cases and in the singular case too with p "not too small," the following estimate holds true 1 for every t ∈ (0, T ), (1.10) where C depends only on α, p and N and (see [28,42,54,55] and the references therein). The previous estimates, known in the literature as decay or ultracontractive estimates or smoothing effects (see [54]), describe the behavior in time of the solutions and are of interest not only for t large, since they describe in which way u decays to zero letting t → +∞, but also for t near zero, since they describe the "controlled blow-up" of the L ∞ -norm of u(t) for t that goes to zero.
The first aim of this paper is to understand what happens when the reaction term f is not the zero function. In other words, we want to understand if for suitably regular data f (otherwise the result is false) it is still true that even in the presence of an only summable initial datum u 0 there exists a solution of (1.1) which becomes "instantaneously bounded" and, in case of an affirmative answer, which kind of estimates we can expect.
The author has already studied this problem in the nonsingular case p ≥ 2 when the operator a is independent of u (i.e., a(x, t, s, ξ) = a(x, t, ξ)) and satisfies the following monotony assumption for a.e. (x, t) ∈ T and for every ξ , η ∈ R N , where γ is a positive constant (see [46] if p = 2 and [47] if p > 2). Hence, we want to complete here our previous studies considering not only the singular case p < 2, but also the more general case of operators depending also on u and satisfying the weak monotone condition (1.5) instead of the strong monotone condition (1.11), essential in the previous papers for the results obtained there.
We recall that by the classical theory we know that even if we have no information on the initial datum u 0 , if f is sufficiently regular (for example, if it is bounded) every solution u belonging to L ∞ loc ((0, T ]; L 2 ( )) ∩ L p loc ((0, T ]; W 1, p 0 ( )) becomes bounded but if u 0 is not regular enough we do not know if we have the starting regularity on u needed to apply these classical results. Moreover, also if we assume more regularity on u 0 , like for example that it is in L 2 ( ), in order to have the required regularity on u, the L ∞ -estimates we get by the classical theory are of different types and don't become the decay estimates presented above when f is the zero function.
Our first result is that even if the initial datum is a summable function, if f is bounded then there exists a solution u of (1.1) that becomes instantaneously bounded (see Sect. 2.1). We prove this regularization phenomenon at first when α 0 > 0 (see Theorem 2.1: the absorption case) and then we show that the immediately boundedness appears also in the absence of the lower-order term α 0 u (see Theorem 2.2: the case without absorption) or even in the presence of the reaction term −α 0 u (see Theorem 2.3).
Moreover, we prove L ∞ -estimates that when f is the zero function become the decay estimates described above.
The second aim of this paper is the study of the behavior in time of solutions that become immediately bounded, even when the initial data are merely summable functions (see Sect. 2.2 for the results). As a matter of fact, every solution of (1.1) can be extended in a global solution u defined in all × (0, +∞) (see Definition 2.2 and Theorem 2.4) and we show that if the lower-order term α 0 u is not identically zero, then all the global solutions of (1.1) that become immediately bounded have the same behavior for t large.
Indeed, we treat not only the case of solutions that become immediately bounded but also solutions that immediately improve their regularity but remain unbounded thus considering a wider class of solutions and of data f .
In all the cases studied here, we prove that if u and v are global solutions that become "immediately suitably regular," where u solves (1.1) and v satisfies the same problem satisfied by u but with initial datum v 0 , i.e., The third aim of the paper, is the study of time behavior of the remaining classes of solutions: the ones that do not improve their regularity with time, this can happen when f and u 0 are merely integrable functions, and the ones for which we have no information about their extra regularity.
We point out that in this context the solutions generally are not unique and hence we have the necessity to choose which solution we want to study. We recall that in order to guarantee uniqueness results further requirements on the solutions are needed, like for example to be an entropy solution or a renormalized solution, and also the operator is generally required to be an operator independent of u, i.e., satisfying for a.e. (x, t) ∈ × (0, +∞) and for every s ∈ R and ξ ∈ R N . Hence, we have restricted this study to the case operators satisfying the previous assumption and we have proved that it is possible to distinguish a class of solutions, "the solutions obtained by approximation," named "sola" (see Definition 2.3), for which we will still be able to describe the behavior in time of the solutions. As observed in [32], "the definition of these solutions is quite natural" and is motivated by the usual way to find solutions of problems with irregular data by an approximating procedure where the irregular data are replaced by more regular functions.
We prove here that if u 0 and v 0 belong to L 1 ( ), then the solutions u and v of, respectively, (1.1) and (1.12), which are obtained by approximation satisfying (1.13).
Moreover, we will also show that the further requirement to be a solution obtained by approximation produces uniqueness despite the fact that the data f and u 0 can be assumed to be only summable functions, thus completing the known uniqueness results on sola that require stronger monotony assumption on the operator than the weak hypothesis (1.5) made here (see Remark 2.9 for further details).
In fact, the assumption (1.14) is also responsible for other interesting phenomena (see Sect. 2.3).
For example, are also unique the solutions that although satisfy only summable initial data then become immediately bounded. Indeed, the uniqueness holds true for a larger class of solutions, eventually unbounded, but which immediately becomes suitably regular (see Theorems 2.13 and 2.14).
We notice that most of the results we prove here remain true replacing the lowerorder term α 0 u in (1.1) with a more general Caratheodory function a 0 (x, t, u, ∇u) satisfying, for example a 0 (x, t, s, ξ)s ≥ α 0 s 2 , α 0 ≥ 0, but to make clearer the role of the lower-order term in the following results we avoid this full generality.
The plan of the paper is the following: in the next section, we state our results in all the details. In Sect. 3, we give some known and new results we need in the proofs. Finally, Sect. 4 is devoted to the proofs of all the results set out in Sect. 2.

Main results
We start showing that in the case of bounded forcing term and integrable initial data there exist solutions that become "instantaneously bounded." Then, in Sect. 2.2 we will describe the behavior in time of all the solutions that become "instantaneously bounded" and finally in Sect. 2.3 we will discuss the case of operator independent of u.
Before stating our results, we recall that by a solution of (1.1) we mean the following.

The case of bounded forcing term and integrable initial data
Besides, ∇u belongs to L p ( × (t, T )) N for every t ∈ (0, T ) and the following estimate holds true

6)
where A(t) is as in (2.4). Finally, if p > 2 also the following estimates hold true Here and throughout the paper, we adopt the convention that 1 +∞ = 0 and we denote with c and c(s,v) (respectively) positive constants and functions depending only on the variable in brackets, that can change from one line to the other.

Remark 2.1. We notice that the assumption p > 2N
N +r 0 is a necessary condition since it is a sharp condition to have solutions that become immediately bounded when f ≡ 0. Moreover, for zero reaction term f the previous estimates become the sharp estimates of the heat equation if p = 2 and those of the p-Laplacian otherwise (see [42,54] and the references therein). Therefore, for these estimates a sort of continuous dependence on the datum f applies.

Remark 2.2.
Notice that by the structure condition (1.3) and the regularity of u proved in the previous theorem it follows that u belongs also to C((0, T ]; L 2 ( )).
Moreover, by the "immediately boundedness" of u and the classical regularity theory we deduce (see [22,33]). In particular, u belongs to C((0, T ]; L r ( )) for every r > 1 arbitrarily fixed and estimate (2.4) holds true for every t ∈ (0, T ).

Remark 2.3.
We point out that when p > 2 the L ∞ -estimate (2.7) together with the gradient estimate (2.8) do not depend on the initial datum u 0 and hence are universal estimates.
The following theorem shows that the presence of the lower-order term α 0 u is not essential in this instantaneous improvement of regularity since the immediately L ∞regularization holds true also if α 0 = 0. Moreover, also in this case it is possible to derive L ∞ -estimates of the solutions depending only on the data.
Besides, ∇u belongs to L p ( × (t, T )) N for every t ∈ (0, T ) and the following estimate holds true where B(t) is as before. Finally, if p > 2 also the following bounds (independent of u 0 ) are satisfied (2.14) Remark 2.4. We point out that the estimates in Theorems 2.1 and 2.2 hold true for every solution constructed as limit of sufficiently regular approximating solutions.

Remark 2.5.
Notice that if f ≡ 0 and p = 2 estimate (2.10) becomes which implies (due to the arbitrariness of ρ) Moreover, if otherwise p = 2 (p > 2N N +r 0 ) and f ≡ 0, by (2.10) and the arbitrariness of ρ we deduce that Finally, if p > 2 and f ≡ 0 (2.13) becomes which implies (due again to the arbitrariness of ρ) the following universal estimate Thus, all the L ∞ -estimates in Theorem 2.2 for zero data f become the sharp decay estimates (in terms of the power of t and of the dependence on u 0 L r 0 ( ) ) known in the literature (see [42,44,54,55] and the references therein).
In conclusion, even if the datum u 0 is not bounded, u becomes bounded also in the presence of a nonzero bounded reaction f and L ∞ -estimates of u hold true depending in a "continuous way on the datum f " (more precisely on the L ∞ -norm of f ). Besides and hence (using again the arbitrariness of ρ) we deduce the following estimate of the L p -norm of the gradient of the solutions of (1 . Moreover, if p > 2 and f ≡ 0 estimate (2.14) implies (again by the arbitrariness of ρ) Notice that estimates (2.17)-(2.19) and (2.12) on ∇u seem not available in the literature except in the case p ≥ 2 for operators a in (1.1) independent of u (i.e., satisfying a(x, t, s, ξ) = a(x, t, ξ)) and satisfying stronger monotony assumptions which are essential in the proofs of the results (see [46,47]).
We conclude this subsection studying the case when in (1.1) the absorption term α 0 u is replaced by a reaction term −α 0 u, i.e., ⎧ ⎨ on . (2.20) The following result shows that what happens is quite similar to the case without absorption. In detail, we have the following result.

Theorem 2.3. (The case with the reaction term
N +r 0 , then there exists a solution u of (2.20) satisfying (2.9). Moreover, for every ρ > α 0 and t ∈ (0, T ) it results with α p and σ as in estimate (2.4).
Besides, ∇u belongs to L p ( × (t, T )) N for every t ∈ (0, T ) and the following estimate holds true Finally, if p > 2 also the following bounds (independent of u 0 ) are satisfied

Time behavior of the solutions
In the previous sections, we have considered solutions of our problem (1.1) in T . In this subsection, we show that there exist solutions in all the set ∞ (global solutions) and we study the behavior in time of these global solutions. To this aim, we give the following definition.

Definition 2.2.
We say that u is a global solution of (1.1) or, equivalently, that u is a solution of ⎧ ⎨ on . (2.26) if u is a solution of (1.1) (according to Definition 2.1) for every arbitrarily fixed T > 0. Moreover, we denote by a "global solution of (1.1) relative to the datum f and to the initial datum v 0 " (v 0 ∈ L r 0 ( ) with r 0 ≥ 1) a global solution v (according to the above definition) of the following problem ⎧ ⎨ on . (2.27) We show now that under the structure assumptions (1.2)-(1.5), even if the data f and u 0 are only summable functions, our problem always admits global solutions.
Moreover, if f is bounded and p not too small, then there exist also solutions which immediately become bounded. In detail, we have the following result.
We observe that generally these global solutions are not unique (see next section for further details). Anyway, we show now that if the lower-order term α 0 u is not zero, all the solutions that become "immediately bounded" have the same asymptotic behavior for t → +∞ and this "asymptotic behavior" is the same also for all the global solutions verifying different initial data. In other words, the solutions for large value of t "forget" their initial data.
More in detail, we have the following result.
Assume α 0 > 0 and that u and v are two global solutions of (1.1) relative to the same datum f ∈ L 1 loc ([0, +∞); L 1 ( )) and to initial data, respectively, u 0 and v 0 , where u 0 and v 0 are functions in L r 0 ( ) (r 0 ≥ 1). If there exist t 0 > 0 and r ≥ 2 such that u and v belong to (2.28) Remark 2.6. It is worth to notice that under the structure assumptions of the previous theorem, if we choose p > 2N N +r 0 , f ∈ L ∞ ( ∞ ) ∩ L 1 ( ∞ ) and we assume also the weak monotone condition (1.5), then by Theorem 2.4 for every choice of initial data u 0 and v 0 in L r 0 ( ) (r 0 ≥ 1) there exist at least a global solution u of (1.1) and a global solution v of (2.27) belonging to L ∞ ( ×(t 0 , +∞))∩ L p (t 0 , +∞; W 1, p 0 ( ))∩ C([t 0 , +∞); L r ( )) for every arbitrarily fixed t 0 > 0 and r ≥ 1.
Hence, by Theorem 2.5, all these regular global solutions have the same asymptotic behavior (for t → +∞) free from their initial data and the convergence (2.28) holds true for every r ≥ 1.
Indeed, if to the assumptions of Theorem 2.5 we add the boundedness of the reaction term f , we obtain that every solution in L p (t 0 , +∞; W belongs also to C([t 0 , +∞); L r ( )) for every r ≥ 1 and satisfies (2.28). In detail, it results.
Assume α 0 > 0 and that u and v are two global solutions of (1.1) relative to the same datum f ∈ L ∞ ( ∞ ) and to initial data, respectively, u 0 and v 0 , where u 0 ) then u and v belong to C([t 0 , +∞); L r ( )) for every r ≥ 1 and (2.28) holds true (for every r ≥ 1).
We observe that even if the global solutions u and v are not locally bounded (and hence the previous results cannot apply) it is still possible to prove that they exhibit the same asymptotic behavior if they are "sufficiently regular" as the following result shows.
Theorem 2.6. Let (1.2)-(1.4) hold true with T = +∞ and h ∈ L p ( ∞ ). Assume α 0 > 0 and that u and v are two global solutions of (1.1) relative to the same datum f and to initial data, respectively, u 0 and v 0 , where u 0 and v 0 are functions in L r 0 ( ) (r 0 ≥ 1). If there exists t 0 > 0 such that u and v belong to L p (t 0 , +∞; W 1, p 0 ( )) ∩ L ∞ (t 0 , +∞; L 2 ( )) and f belongs to L p loc ([t 0 , +∞); W −1, p ( )), then u and v belong to C loc ([t 0 , +∞); L 2 ( )) and it results Remark 2.7. We observe that the study of the smoothing effect and time behavior of the solutions to special classes of operators of p-Laplacian type in the absence of reaction terms and in different geometric settings can be found in [13][14][15][16]19,28].
We conclude this section studying the particular case of operator a independent of u since in this case many other further interesting properties of the solutions appear.

The case of operator a independent of u
In this subsection, we prove that under the following condition a(x, t, s, ξ) = a(x, t, ξ) for every (s, ξ) ∈ R × R N and a.e. (x, t) ∈ T , (2.29) many interesting further properties of the solutions of (1.1) appear concerning not only their behavior in time but also their uniqueness properties.
As a first result, we prove that the presence of a bounded reaction term f allows to take r = +∞ in (2.28). More in detail, we have the following result.
Theorem 2.7. Assume that (1.2)-(1.7) and (2.29) hold true with T = +∞ and α 0 > 0. Let u and v be two global solutions of (1.1) relative to the same datum f and to initial data, respectively, u 0 and v 0 , where u 0 and v 0 are functions in L r 0 ( ) (r 0 ≥ 1).
Moreover, the following estimate holds true We point out that in the special autonomous case (for a.e. x ∈ and t ≥ 0 and for all s ∈ R and ξ ∈ R N ) the previous theorem implies that for large value of t the "suitably regular" global solutions behave as the solution Moreover, it is also possible to estimate the difference between these global solutions of (1.1) and the stationary solution w. In detail, we have the following result.
where w is the unique solution of (2.33) belonging to W Remark 2.8. We recall that analogous results where proved in [46,47] when p ≥ 2 and under stronger monotony assumptions on the operator a(x, ξ). Hence, the previous result completes the cited ones revealing that the same phenomenon appears in the singular case p < 2 and for a more general class of "weak monotone" parabolic equations.
Until now, we have described the behavior of suitably regular solutions considering different "type of regularity" (for example, immediately boundedness or gradient in L p ..).
We show now that it is possible to say something also when the solutions generally do not increase their regularity, as for example when f and u 0 are only summable functions, or when we do not have any information on the regularity of the solutions.
As a matter of fact, in all these cases it is possible distinguish a class of solutions, "the solutions obtained by approximation," for which we will still be able to describe their behavior in time. Moreover, we will also show that the further requirement to be a solution obtained by approximation produces uniqueness despite the fact that the data f and u 0 can be assumed to be only summable functions.
More in detail, let us define what we mean by a "solution constructed by approximation." where f n and u 0,n satisfy Our first result on "sola" is the following. and the following "estimate of continuous dependence from the data" holds true for every t ∈ (0, T ) where v is the (unique) solution constructed by approximation of problem (1.1) which assume initial datum v 0 ∈ L 1 ( ) and forcing term g ∈ L 1 ( T ), i.e., v is the solution constructed by approximation of the following problem ⎧ ⎨ (2.40) Remark 2.9. We notice explicitly that in Theorem 2.8 we have assumed α 0 ≥ 0 and hence the uniqueness of sola holds true also in the absence of the lower-order term α 0 u. We also point out that the previous uniqueness result completes the result in [20] where when p > 2 − 1 N +1 the uniqueness of solutions obtained by approximation is proved assuming the following further structural assumptions We prove now that if the forcing term f is bounded and p is not too small then the unique solution of (1.1) obtained by approximation is a solution which "immediately becomes bounded." In detail, we have Indeed, by the following theorem we can affirm that the solutions obtained by approximations have also a unique extension to a global solution obtained by approximation and all these global solutions exhibit the same behavior for large values of t, independently from the initial data that they assume. We recall that here by a global solution defined by approximation we mean the following is the global solution obtained by approximation of (2.40) with g ∈ L 1 loc ([0, +∞); L 1 ( )) and v 0 ∈ L r 0 ( ), then the estimate of continuous dependence from the data (2.39) holds true for every t > 0. Finally, if α 0 > 0 and g = f (hence now v is the unique global solution obtained by approximation of (2.27)) then the following estimate holds true for a.e. t > 0 where c = | | Remark 2.10. We point out explicitly that by (2.42) it follows that all these global solutions obtained by approximation have the same asymptotic behavior at infinity even if they satisfy different initial data.
We show now that if p is not too small and in the presence of a more regular datum f the unique global solution of (1.1) obtained by approximation is a solution which "immediately increases its regularity." In detail, we have for every t 0 > 0 and r ≥ 1.
Indeed, even if a solution is not obtained by approximation, if it is sufficiently regular (belonging to L p loc ([0, +∞); W 1, p 0 ( ))), then it has the same asymptotic behavior for t → +∞ of the solutions constructed by approximation. As a matter of fact, it results.
In particular, the limit (2.42) is satisfied.
Indeed, the previous result reveals that if α 0 > 0 then all the "sufficiently" regular solutions are unique and coincide with the solutions obtained by approximations. As a matter of fact, an immediate consequence of estimate (2.43) (applied with u 0 = v 0 ) is the following result. We notice that in the previous uniqueness result it was essential the requirement that α 0 > 0, otherwise the crucial estimate (2.43) fails. We prove now that the uniqueness of the "sufficiently regular" solutions holds true even if α 0 = 0.
Moreover, we prove that it is sufficient that the solution becomes immediately "sufficiently regular" (i.e., it is not necessary the regularity "up to zero"). In detail, we have the following result.  Moreover, if we also assume α 0 > 0 and f ∈ L ∞ ( ∞ ) ∩ L 1 ( ∞ ), then u belongs also to for every t 0 > 0 and r ≥ 1. Finally, u is also the unique global solution obtained by approximation. Remark 2.11. We recall, that we have proved above that under the assumptions of the previous theorem there exists a global solution which immediately increases its regularity which is the unique solution obtained by approximation. A priori we do not know if there exist other global solutions which immediately increases their regularity and that are not solutions obtained by approximation. The previous result allows to conclude that there exists one and only one global solution which immediately become regular. Thus, such a unique global solution which becomes immediately regular is also the unique global solution obtained by approximation.

Preliminary results
In this section, we give some known and new results which will be essential in the proofs of the results presented here and that can be of interest in themselves.
We start recalling some results about L ∞ -decay estimates proved in [42] which substantially allow to obtain decays estimates simply by suitable integral estimates (see [48] for a brief description of this method). The proofs of these results make use of a technical Lemma proved in [50] together with a classical Lemma (see Lemma 3.1) often used to prove L ∞ -results. We point out that there are different methods to prove decay estimates. For example, a classical method (completely different from that used here) is the use of suitable Sobolev logarithmic inequalities which reflect the operator involved in the problem (see for example [19,27,54] and the references therein). In detail, let us define G k (s) = (|s| − k) + sign(s). (3.1) We have the following results.
where is an open set of R N , (not necessary bounded), N ≥ 1, 0 < T ≤ +∞ and Suppose that u satisfies the following integral estimates for every k > 0

5)
where c 1 and c 2 are positive constants independent of k. Finally, let us define Then, there exists a positive constant C 1 (see Formula (4.11) in [42]) depending only on N , c 1 , c 2 , r, r 0 , q and b such that

7)
where Moreover, if has finite measure we have an exponential decay if b = r and universal bounds if b > r . More in detail, we have the following result. [42]) Let the assumptions of Theorem 3.1 hold true. If has finite measure and b = r , the following exponential decay occurs

10)
and where | | denotes the measure of . If otherwise has finite measure and b > r, we have the following universal bound

12)
where (3.13) and C (see Formula (4.19) in [42]) is a constant depending only on r , r 0 , q, b, c 1 , c 2 and the measure of .
Remark 3.1. The previous Theorems hold true also if we replace assumption (3.5) with the following weaker assumption (3.14) and in this last case all the previous decay estimates hold true with the value u 0 L r 0 ( ) replaced by c 0 c 2 . For details, see Remark 2.4 in [42]. The following three theorems are generalizations of Theorems 3.1 and 3.2 and will be proved at the end of this section.
where u 0 is as in (3.6), the exponents h 0 and h 1 are as in (3.8) and C 0 = max{2, C 1 } with C 1 as in Theorem 3.1.

Theorem 3.4. Under the assumptions of Theorem 3.3, if has finite measure and b = r then the following estimate occurs
where u 0 is as in (3.6), the exponent h 1 is as in (3.10), σ is as in (3.11) and C 3 = max{2, C 2 } (C 2 as in Theorem 3.2).
for every t ∈ (0, T ), (3.17) where the exponent h 2 is as in (3.13) and C 4 = max{2, C } is a constant independent of u 0 (C is as in Theorem 3.2).
In the proofs of Theorems 3.3-3.5 below, we will use the following lemma which, as recalled above, is a very useful tool in proving boundedness results. and where C and δ are positive given constants and B ≥ 1. Then, it follows that To be self-contained, we recall also the following two results, proved, respectively, in [38,43], which we will use in the proofs of the asymptotic behavior of the solutions. Theorem 3.6. (Theorem 2.8 of [43]) Let u be in C((0, T ); L r ( )) ∩ L ∞ (0, T ; L r 0 ( )) where 0 < r ≤ r 0 < +∞. Suppose also that | | < +∞ if r = r 0 (no assumption on | | are needed if r = r 0 ). If u satisfies for every 0 < t 1 < t 2 < T, (3.20) and there exists u 0 ∈ L r 0 ( ) such that u(t) L r 0 ( ) ≤ c 2 u 0 L r 0 ( ) for almost every t ∈ (0, T ), (3.21) where c i (i=1,2) are real positive constants, then the following estimate holds true

is a positive constant and g is a nonnegative function belonging to L
We conclude this section by proving Theorems 3.3-3.5.

Proof of Theorem 3.3.
Proceeding exactly as in the proof of the Theorem 2.1 in [42] (recalled above as Theorem 3.1), we deduce that the following inequality holds (see (4.7) in [42]) Notice that δ is a positive constant being, by assumption, r > r 0 and b 0 < b < q. Since (3.24) is obtained applying the integral inequality (3.4) and now such estimate is true only for k ≥ σ 0 , we need to require that k m ≥ σ 0 for every m ∈ N ∪ {0} and hence, being k 0 = k 2 = min{k m }, this is true if k ≥ 2σ 0 . To conclude the proof, we need to show that also the assumption (3.19) of Lemma 3.1 is satisfied, i.e., that it results Now proceeding exactly as that of Theorem 2.1 in [42] but with the further requirement that k has to be chosen satisfying also the inequality k ≥ 2σ 0 we obtain from which the assertion follows.
Proof of Theorem 3.4. Analogously to the proof of the previous theorem, it is sufficient to proceed exactly as in the proof of Theorem 2.2 (case b = r ) in [42] (recalled above as Theorem 3.2) but with the further requirement that k has to be chosen satisfying also the inequality k ≥ 2σ 0 . This restriction on k implies that the following estimate holds true (instead of (3.9)) where C 2 , σ and h 1 are as in Theorem 3.2. From the last inequality, the assertion follows.
Proof of Theorem 3.5. Analogously to the proof of the previous theorem, it is sufficient to proceed exactly as in the proof of Theorem 2.2 (case b > r ) in [42] but once again with the further requirement that k has to be chosen satisfying also the inequality k ≥ 2σ 0 . This restriction on k implies that the following estimate holds true (instead of (3.12)) where C and h 2 are as in Theorem 3.2. From the last inequality, the assertion follows.

Proofs
Proof of Theorem 2.1. Let u be the solution constructed as the a.e. limit in T of the solutions u n ∈ L ∞ ( T ) ∩ L p (0, T ; W 1, p 0 ( )) ∩ C( T ) to the following approximating problems ⎧ ⎨ ⎩ (u n ) t − div(a(x, t, u n , ∇u n )) + α 0 u n = f (x, t) in T , u n = 0 o n , u n (x, 0) = u 0,n (x) on , where the data u 0,n (x) ∈ C ∞ c ( ) satisfy (see for example [7,8,44]). The proof proceeds in two steps. Firstly, we prove estimates (2.4) and (2.7) and then we show that also estimates (2.6) and (2.8) hold true.
Step 1. To prove estimate (2.4), it is sufficient to demonstrate that the following inequality holds where A(t) is as in (2.4). Let ε and k be positive constants arbitrarily fixed and take ϕ = {[ε+(|u n |−k) + ] r −1 −ε r −1 }sign(u n ) as test function in (4.1) where r > r 0 (hence r > 1) is arbitrarily fixed too. Notice that the use of such a test function, together with the following ones, can be made rigorous by means of Steklov averaging process.
Using assumption (1.2), we obtain for every 0 ≤ t where G k is the function defined in (3.1) and Applying Sobolev 3 inequality in the previous estimate, we deduce where c S = c S ( p, N ). Notice that it results c S < p * p = (N −1) p N − p see [18]. where the last integral in the previous inequality is nonnegative. Hence, letting ε → 0 we deduce that for every 0 ≤ t 1 < t 2 ≤ T and k ≥ σ 0 the following inequality holds where we have denoted We recall that we have chosen r > r 0 . Moreover, since it results that is, again true by our assumption on p. Thus, the previous choice of b, r and q satisfy (3.3). Observe that if r 0 > 1, proceeding as above but with r = r 0 we deduce that (4.5) holds true also with r = r 0 , and thus the following estimate follows and hence (3.5) is satisfied. Otherwise, if r 0 = 1, taking and using (1.2) (and that δ > 1) we obtain for every 0 ≤ t 0 < t < T from which (letting δ → +∞) we deduce for every 0 ≤ t 0 ≤ t < T and k ≥ σ 0 Notice that if p > 2 it results b > r . Hence, applying Theorem 3.5 and noticing that b − r = p − 2 it follows that which implies the bound (2.7).
Step 2. As said above, we prove here that also estimates (2.6) and (2.8) hold. To this aim, taking u n as test function in the approximating problem (4.1) we obtain for every 0 < t ≤ T We estimate now the right-hand side of the previous inequality. By (4.3), it follows Notice that again by (4.3) it results Hence, by the previous two inequalities we deduce By (4.10)-(4.12), we conclude that (4.13) from which the assertion (2.6) follows. Finally, if p > 2, proceeding exactly as above but using (4.9) instead of (4.3) and hence replacing A(t) with U (t) (see (2.7)) we deduce that also the following estimate holds true from which (2.8) follows.
Proof of Theorem 2.2. Let ρ > 0 arbitrarily fixed and consider the following nonlinear on , (4.14) where we have defined where h is as in (1.4) and (4.20) Moreover, by assumption (1.7) we deduce that f belongs to L ∞ ( T ). Hence, we can apply Theorem 2.1 with α 0 = ρ (see also Remark 2.2) and conclude that there exists a solution v of (4.14) belonging to C( ×(0, T ]) satisfying (for every t ∈ (0, T )) the following estimates where α is as in (4.19), σ = αc S c(r 0 ) is the Sobolev constant defined in (4.4)) and the exponents h 0 and h 1 are as in (2.5). Moreover, ∇v ∈ L p ( × (t, T )) N for every t ∈ (0, T ) and also the following estimate holds true where A(t) is as in (4.21). Besides, if p > 2 also the following estimates are satisfied Notice that the function u = ve ρt is a solution of (1.1). Hence, we can rewrite (4.21)-(4.24) as Now estimates (2.10) follow by (4.25) while estimate (4.27) implies estimate (2.13).
Notice since it results by estimates (4.26) and (4.28) it is possible to derive the following estimates for ∇u (4.29) and if p > 2 where B(t) and B 0 are as in the statement of the theorem.
Indeed, once we know that estimates (2.10) and (2.13) are satisfied it is possible to derive a slightly better estimate on ∇u, respect to estimates (4.29) and (4.30) obtained above, using directly the equation satisfied by u instead of the estimates of ∇v. As a matter of fact, it results (recalling the construction of u) for every 0 < t < T Now, estimate (2.12) is an immediate consequence of estimates (4.32) and (2.10) while (2.14) follows by (4.32) and (2.13) .
Proof of Theorem 2.3. As in the proof of the previous theorem, we consider an auxiliary problem. In detail, let ρ > α 0 and let us consider the following problem where a and f are as in (4.15). Now it is sufficient to proceed exactly as in the proof of Theorem 2.2 to get the result with the only difference that in applying Theorem 2.1 the absorption term is now (ρ − α 0 )v instead ρv. Notice that this change in the absorption term is the reason of the slow differences in the bounds satisfied by the solution u of (2.20) respect to the bounds satisfied by the solution described in Theorem 2.2 (i.e., the when α 0 = 0).

Remark 4.1.
We notice that the estimates of the previous theorems hold true also replacing assumption (1.4) on h with the weaker hypothesis h ∈ L p ( T ).
Proof of Theorem 2.4. The proof proceeds in two steps. Firstly, we prove the existence of a global solution u when f ∈ L 1 loc ([0, +∞); L 1 ( )) and the regularity C loc ([0, +∞); L 1 ( )) when the operator a is independent of u. Then, in Step 2, we show the further regularity properties of u when p > 2N N +r 0 . Step 1. Let us consider the following approximating problems where u 0,n ∈ C ∞ c ( ) : u 0,n → u 0 in L 1 ( ) (4.35) and f n (x, t) ∈ L ∞ loc ([0, +∞); L ∞ ( )) satisfy for every T > 0 : We recall that for every fixed n ∈ N, there exists a unique global solution u n of (4.34) in Moreover, for every arbitrarily fixed T > 0, there exists a subsequence of the sequence u n converging a.e. in T (indeed a stronger convergence holds) to a solution of our problem (1.1) in T (see for example [6][7][8][9]44,52]).
Using the above properties of the sequence u n we construct now a global solution u of (1.1). To this aim, let T 0 be a positive constant arbitrarily fixed. As said above, there exists a subsequence u (1) n of u n satisfying u (1) n → u 1 a.e. in T 0 where u 1 is a solution of (1.1) in T 0 . We know that every term of the sequence u (1) n is a global solution, and therefore it is also a solution in 2T 0 . Hence, for what recalled above, there exists a subsequence u (2) n of u (1) n satisfying where u 2 is a solution of (1.1) in 2T 0 . Iterating this procedure, we can define u (in all the set × (0, +∞)) in the following way where h ∈ N is such that hT 0 > T . We point out that u is well defined since (by construction) if m ∈ N is such that mT 0 > T then it results We point out that u is a global solution of (1.1) since (by construction) for every arbitrarily fixed T > 0 u solves (1.1).
To conclude this step, it remains to show that if a(x, t, s, ξ) ≡ a(x, t, ξ) then u belongs to C loc ([0, +∞); L 1 ( )). Notice that it is sufficient to show that for every T > 0 arbitrarily fixed the sequence u n is a Cauchy sequence in C([0, T ]; L 1 ( )). The proof of this last fact is similar at all to that in [52] (see also [49]) and hence we omit it.
Step 2. Assume p > 2N N +r 0 . We distinguish two cases: the "more regular case" (case 1) with stronger assumptions on f and α 0 > 0 and the "less regular case" (case 2) with a weaker assumption on f and α 0 ≥ 0. Case 1: Let f be in L ∞ ( ∞ )∩ L 1 ( ∞ ) and α 0 > 0. Let u be the solution constructed in the previous step. Thanks to the further assumptions, we have done here we can choose in the above approximating problems (4.34) and u 0,n (x) satisfying u 0,n (x) → u 0 in L r 0 ( ), u 0,n (x) L r 0 ( ) ≤ u 0 (x) L r 0 ( ) . (4.38) Thanks to the above construction of u and recalling the assumption α 0 > 0, for every T > 0 arbitrarily fixed we can proceed exactly as in the proof of Theorem 2.1 obtaining that u satisfies estimates (2.4) and (2.6) for every T > 0 arbitrarily fixed. Consequently, for every t 0 > 0 it follows and +∞ t 0 where α is as in (1.2), c S = c S (N ) is the Sobolev constant defined in (4.4) and σ , h 0 and h 1 are as in (2.4). Hence, u belongs to L ∞ ( × [t 0 , +∞)) ∩ L p (t 0 , +∞; W 1, p 0 ( )) for every arbitrarily fixed t 0 > 0. To conclude the proof, it remains to show that u belongs also to C([t 0 , +∞); L r ( )) for every r ≥ 1 arbitrarily chosen. To this aim, we observe that by the structure assumption (1.3), the regularity assumed on f and the above regularity proved on u imply that u belongs also to C([t 0 , +∞); L 2 ( )). Hence, by the classical theory (see [22,33]) we obtain that u is in C( × [t 0 , +∞)). Thus, it follows that u belongs also to C([t 0 , +∞); L r ( )) for every r ≥ 1. Case 2: Let f be in L ∞ loc ([0, +∞); L ∞ ( )) and α 0 ≥ 0. Thanks to the further regularity assumption on f , we have done here we can choose in the approximating problems (4.34) f n and u 0,n (x) as in the previous case, i.e., satisfying (4.37) and (4.38). Thanks to the above construction of u and recalling the assumption on p done in this step, for every T > 0 arbitrarily fixed we can proceed exactly as in the proof of Theorem 2.2 obtaining that u satisfies estimates (2.10) and (2.12) for every T > 0 arbitrarily fixed. Thus, u belongs to L ∞ loc ((0, +∞); L ∞ ( )) ∩ L p loc ((0, +∞); W 1, p 0 ( )). Now the remaining regularity C loc ((0, +∞); L r ( )) can be proved reasoning as in Case 1 and hence we omit it.
Moreover, if in addition we assume that u and v belong to L ∞ (t 1 , +∞; L 2 ( )) for some t 1 ≥ t 0 , then for every t > t 2 ≥ t 1 arbitrarily fixed it results |u − v| 2 (t) ≤ 3 e −2α 0 (t−t 2 ) + Proof of Theorem 4.2. We observe that by the structure assumption (1.3) and by the regularity assumption done on u, v and f it follows that u and v belong to C loc ([t 0 , +∞); L 2 ( )). Now the assertions of the theorem follow choosing r = 2 and proceeding exactly as in the proof of Theorem 4.1.
Proof of Corollary 2.1. The proof that both the solutions u and v belong to C([t 0 , +∞); L r ( )) for every r ≥ 1 is at all similar to that in the proof of Theorem 2.4. In detail, by the structure assumptions and the regularity assumed on u, v and f it follows that the solutions u and v belong to C([t 0 , +∞); L 2 ( )) and hence, by the classical theory (see [22,33]) they are also in C( × [t 0 , +∞)). Thus, the regularity C([t 0 , +∞); L r ( )) (for every r ≥ 1) follows and consequently estimate (2.28) holds true by Theorem 2.5 for every r ≥ 2 (and hence for every r ≥ 1).
Proof of Theorem 2.6. The assertion follows by Theorem 4.2.
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