Global existence of solutions and smoothing effects for classes of reaction–diffusion equations on manifolds

We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^n$$\end{document}.


Introduction
We investigate existence of global in time solutions to nonlinear reaction-diffusion problems of the following type: where M is an N -dimensional complete noncompact Riemannian manifold of infinite volume, being the Laplace-Beltrami operator on M and T ∈ (0, ∞]. We shall assume throughout this paper that so that we are concerned with the case of degenerate diffusions of porous medium type (see [40]), and that the initial datum u 0 is nonnegative. Let L q (M) be the space of those measurable functions f such that | f | q is integrable w.r.t. the Riemannian measure μ. We shall always assume that M supports the Sobolev inequality, namely that: where C s is a positive constant and 2 * := 2N N −2 . In one of our main results, we shall also suppose that M supports the Poincaré inequality, namely that: for some C p > 0. Observe that, for instance, (1.2) holds if M is a Cartan-Hadamard manifold, i.e. a simply connected Riemannian manifold with nonpositive sectional curvatures, while (1.3) is valid when M is a Cartan-Hadamard manifold satisfying the additional condition of having sectional curvatures bounded above by a constant −c < 0 (see, e.g. [11,12]). Therefore, as is well known, in R N (1.2) holds, but (1.3) fails, whereas on the hyperbolic space both (1.2) and (1.3) are fulfilled.

On some existing results
In [14], problem (1.1) has been studied when p < m. We refer the reader to such paper for a comprehensive account of the literature; here we limit ourselves to recall some results particularly related to ours. For M = R N and m = 1, it is well known that, if p ≤ 1 + 2 N , then the solution of problem (1.1) blows up in finite time for any u 0 ≡ 0, while global existence holds if p > 1 + 2 N and u 0 is bounded and small enough (see [8,22]; for further results, see also [7,9,10,25,32,35,36,39,44,45]). For m > 1, in [38] it is shown that the solution to problem (1.1) blows up for any p ≤ m + 2 N , u 0 ≡ 0; instead, there exists a global in time solution provided p > m + 2 N and u 0 is compactly supported and sufficiently small. On Riemannian manifolds satisfying suitable volume growth conditions, for m = 1 and p ≤ 1 + 2 N , in [29,46] it is proved that the solution of problem (1.1) blows up for any u 0 ≡ 0, while global existence holds if p > 1 + 2 N for small enough initial data u 0 . Similar results have also been stablished in [5,34,42,43].
Problem (1.1), without the forcing term u p , has been largely studied on Riemannian manifolds, and in particular on Cartan-Hadamard manifolds, in [6,13,15,16,18,19,21,33,41]. In [20] problem (1.1) is addressed on Cartan-Hadamard manifolds with −k 1 ≤ sec ≤ −k 2 for some k 1 > k 2 > 0, where sec denotes the sectional curvature. It is shown that, for any p > m, there exists a global in time solution, provided that u 0 has compact support and is small enough, while if u 0 is large enough, then there exists a solution blowing up in finite time.
For any x 0 ∈ M, r > 0, let B r (x 0 ) be the geodesic ball centred in x 0 and radius r , let g i j the metric tensor. In [46], problem (1.1) is studied when M is a manifold with a pole, μ(B r (x 0 )) ≤ Cr α for some α > 2 and C > 0. Under an additional smallness condition on curvature at infinity, if u 0 is sufficiently small and with compact support, then there exists a global solution to problem (1.1). Global existence is also proved, for some initial data u 0 , under the assumption that M has nonnegative Ricci curvature and p > α α−2 m. It should be noticed that such result does not cover cases in which negative curvature either does not tend to zero at infinity, or does so not sufficiently fast, in particular the case of the hyperbolic space cannot be addressed. Finally, in [14] global existence of solutions to problem (1.1) is obtained, for any p < m and u 0 ∈ L m (M), under the assumption that the Sobolev and the Poincaré inequalities hold on M.

Qualitative statements of our new results in the Riemannian setting
Our results concerning problem (1.1) can be summarized as follows.
• (See Theorem 2.2) We prove global existence of solutions to (1.1), assuming that the initial datum is sufficiently small, that and that the Sobolev inequality (1.2) holds; moreover, smoothing effects and the fact that suitable L q norms of solutions decrease in time are obtained. To be specific, any sufficiently small initial datum u 0 ∈ L m (M) ∩ L ( p−m) N 2 (M) gives rise to a global solution u(t) such that u(t) ∈ L ∞ (M) for all t > 0 with a quantitative bound on the L ∞ norm of the solution. for any sufficiently small initial datum u 0 , belonging to suitable Lebesgue spaces, there exists a global solution u(t) such that u(t) ∈ L ∞ (M). Furthermore, a quantitative bound for the L ∞ norm of the solution is satisfied for all t > 0.
Note that in Theorem 2.2 we only assume the Sobolev inequality and we require that p > m + 2 N , instead in Theorem 2.5 we can relax the assumption on the exponent p, indeed we assume p > m, but we need to further require that the Poincaré inequality holds. Moreover, in the two theorems, the hypotheses on the initial data are different.
The main results given in Theorems 2.2 and 2.5 depend essentially only on the validity of inequalities (1.2) and (1.3), are functional analytic in character and hence can be generalized to different contexts.

The case of Euclidean, weighted diffusion
As a particularly significant setting, we single out the case of Euclidean, massweighted reaction-diffusion equations, that has been the object of intense research. In fact, we consider the problem where ρ : R N → R is strictly positive, continuous and bounded, and represents a mass variable density . The problem is naturally posed in the weighted spaces This kind of problem arises in a physical model provided in [23]. Such choice of ρ ensures that the following analogue of (1.2) holds: for a suitable positive constant C s . In some cases, we also assume that the weighted Poincaré inequality is valid, that is for some C p > 0. For example, (1.6) is fulfilled when ρ(x) |x| −a , as |x| → +∞, for every a ≥ 2, whereas, (1.5) is valid for every a > 0. Problem (1.4) under the assumption 1 < p < m has been investigated in [14]. Under the assumption that the Poincaré inequality is valid on M, it is shown that global existence and a smoothing effect for small L m initial data hold, that is solutions corresponding to such data are bounded for all positive times with a quantitative bound on their L ∞ norm.
In [26,27], problem (1.4) is also investigated, under certain conditions on ρ. It is proved that if ρ(x) = |x| −a with a ∈ (0, 2), and u 0 ≥ 0 is small enough, then a global solution exists (see [26,Theorem 1]). Note that the homogeneity of the weight ρ(x) = |x| −a is essentially used in the proof, since the Caffarelli-Kohn-Nirenberg estimate is exploited, which requires such a type of weight. In addition, a smoothing estimate holds. On the other hand, any nonnegative solution blows up, in a suitable sense, when ρ(x) = |x| −a or ρ(x) = (1 + |x|) −a with a ∈ [0, 2), u 0 ≡ 0 and Furthermore, in [27,28], such results have been extended to more general initial data, decaying at infinity with a certain rate (see [27]). Finally, in [26,Theorem 2], it is shown that if p > m, ρ(x) = (1 + |x|) −a with a > 2, and u 0 is small enough, a global solution exists. Problem (1.4) has also been studied in [30,31], by means of suitable barriers, supposing that the initial datum is continuous and with compact support. In particular, in [30] the case that ρ(x) |x| −a for |x| → +∞ with a ∈ (0, 2) is addressed. It is proved that for any p > 1, if u 0 is large enough, then the solution blows up in finite time. On the other hand, if p >p, for a certainp > m depending on m, p and ρ, and u 0 is small enough, then there exists a global bounded solution. Moreover, in [31] the case that a ≥ 2 is investigated. For a = 2, blow-up is shown to occur when u 0 is big enough, whereas global existence holds when u 0 is small enough. For a > 2, it is proved that if p > m, u 0 ∈ L ∞ loc (R N ) and goes to 0 at infinity with a suitable rate, then there exists a global bounded solution. Furthermore, for the same initial datum u 0 , if 1 < p < m, then there exists a global solution, which could blow up as t → +∞ .
Our main results concerning problem (1.4) can be summarized as follows. Assume • (See Theorem 2.8) We prove that (1.4) admits a global solution, provided that moreover, certain smoothing effects for solutions are fulfilled. More precisely, for any sufficiently small initial datum for all t > 0 and a quantitative bound on the L ∞ norm is verified. Moreover, suitable L q norms of solutions decrease in time.
• (See Theorem 2.9) We show that, if the Poincaré inequality (1.6) holds and one assumes the condition p > m, then, for any sufficiently small initial datum u 0 belonging to suitable Lebesgue spaces, there exists a global solution u(t) to (1.4) such that u(t) ∈ L ∞ (R N ), with a quantitative bound on the L ∞ norm.
Let us compare our results with those in [26]. Theorem 2.8 deals with a different class of weights ρ with respect to [26,Theorem 1], where ρ(x) = |x| −a for a ∈ (0, 2), and the homogeneity of ρ is used. As a consequence, also the hypotheses on p and the methods of proofs are different. Furthermore, Theorem 2.9 requires the validity of the Poincaré inequality, hence, in particular, it can be applied when ρ(x) = (1 + |x|) −a with a ≥ 2 (see [17]). On the other hand, in Theorem [26,Theorem 2] it is assumed that ρ(x) = (1 + |x|) −a for a > 2, so, the case a = 2 is not included.

Organization of the paper
In Section 2, we state all our main results. In Section 3, some auxiliary results concerning elliptic problems are deduced together with a Benilan-Crandall-type estimate. In Section 4, we introduce a family of approximating problems. Then, for such solutions, we prove that suitable L q norms of solutions decrease in time, and a smoothing estimate, in the case p > m + 2 N , supposing that M supports the Sobolev inequality. Under such assumptions, global existence for problem (1.1) is shown in Section 5. In Section 6, we prove that suitable L q norms of solutions decrease in time, and L ∞ bounds for solutions of the approximating problems, under the assumptions that p > m and that M supports the Poincaré inequality as well. Then, under such hypotheses, existence of global solutions to problem (1.1) is proved. Finally, a concise proof of the results concerning problem (1.4) is given in Section 7 by adapting the previous methods to that situation.

Statements of main results
We state first our results concerning solutions to problem (1.1), then we pass to the ones valid for solutions to problem (1.4).

Global existence on Riemannian manifolds
Solutions to (1.1) will be meant in the very weak, or distributional, sense, according to the following definition.
First, we consider the case that p > m + 2 N and the Sobolev inequality holds on M. In order to state our results, we define Assume that with ε 0 = ε 0 ( p, m, N , r, C s ) sufficiently small. Then, problem (1.1) admits a solution for any T > 0, in the sense of Definition 2.1. Moreover, for any τ > 0, one has u ∈ L ∞ (M × (τ, +∞)) and there exists a numerical constant > 0 such that, for all t > 0, one has Moreover, let p 0 ≤ q < ∞ and Finally, for any with ε = ε( p, m, N , r, C s , q) sufficiently small, then We notice that the proof of the above theorem will show that one can take an explicit value of ε 0 in (2.2). In fact, let q 0 > 1 be fixed and {q n } n∈N be the sequence defined by: so that (2.7) Clearly, {q n } is increasing and q n −→ +∞ as n → +∞. Fix q ∈ [q 0 , +∞) and let n be the first index such that qn ≥ q. Definẽ Observe that ε 0 in (2.8) depends on the value of q through the sequence {q n }. More precisely,n is increasing with respect to q, while the quantity min n=0,...,n 2m(q n −1) We then let q 0 = p 0 , take q = pr and define, for these choice of q 0 , q, Similarly, one can choose the following explicit value for ε in (2.5): N and u 0 has compact support and is small enough, then the solution to problem (1.1) globally exists and decays like Note that under these assumptions, Theorem 2.2 can be applied. It implies that the solution to problem (1.1) globally exists and decays like It is easily seen that, for any p ≥ m 1 Hence, when p ≥ m 1 + 2 N the decay's rate of the solution u(t), for large times, given by Theorem 2.2 is better than that of [38, Theorem 3, p. 220], while the opposite is true for m + 2 N < p < m 1 + 2 N . In both cases, the class of initial data considered in Theorem 2.2 is wider.
In the next theorem, we address the case that p > m, supposing that both the inequalities (1.2) and (1.3) hold on M.

Theorem 2.5. Let M be a complete, noncompact manifold of infinite volume such that the Sobolev inequality (1.2) and the Poincaré inequality
Assume that ) and for all t > 0 one has for some ε 2 = ε 2 ( p, m, N , r, C p , C s , q) sufficiently small. Then, Remark 2.6. We define, given q > 1: . The proof will show that one can choose Similarly, we observe that in (2.12) we can choose In the next sections, we always keep the notation as in Remarks 2.3 and 2.6.

Weighted, Euclidean reaction-diffusion problems
We consider a weight ρ : Solutions to problem (1.4) are meant according to the following definition.
Let the weight ρ satisfy (2.16). We say that the function u is a solution to problem (1.4) First, we consider the case that p > m + 2 N . Recall that since ρ is bounded, the Sobolev inequality (1.5) necessarily holds.
Assume that ) admits a solution for any T > 0, in the sense of Definition 2.7. Moreover, for any τ > 0, one has u ∈ L ∞ (R N × (τ, +∞)) and there exist > 0 such that, for all t > 0, one has Moreover, let p 0 ≤ q < ∞ and A quantitative form of the smallness condition on u 0 in the above theorem can be given exactly as in Remark 2.3, see in particular (2.8), (2.9) and (2.10).
In the next theorem, we address the case p > m. We suppose that the Poincaré inequality (1.6) holds. Theorem 2.9. Let ρ satisfy (2.16) and assume that the inequality (1.6) hold. Let admits a solution for any T > 0, in the sense of Definition 2.7. Moreover, for any τ > 0 one has u ∈ L ∞ (R N × (τ, +∞)) and for all t > 0 one has for some ε 2 = ε 2 ( p, m, N , r, C p , C s , q) small enough. Then, A quantitative form of the smallness condition on u 0 in the above theorem can be given exactly as in Remark 2.6, see in particular (2.14) and (2.15).

Auxiliary results for elliptic problems
with centre x 0 ∈ M and radius R > 0. If a reference point x 0 ∈ M is fixed, we shall simply denote by B R the ball with centre x 0 and radius R. Moreover, we denote by μ the Riemannian measure on M.
For any given function v, we define for any k ∈ R + where u 0 ∈ L ∞ (B R ), u 0 ≥ 0. Solutions to problem (3.2) are meant in the weak sense as follows.
We also consider elliptic problems of the type In the next lemma, we recall [14, Lemma 3.6], which will be used later.
The following proposition contains an estimate in the spirit of the L ∞ one of Stampacchia (see, e.g. [4,24] and references therein) in the ball B R ; however, some differences are in order. In fact, we aim at obtaining an estimate independent of the radius R (see Remark 3.5). Since the volume of M is infinite, the classical estimate of Stampacchia cannot be directly applied.
Remark 3.5. If in Proposition 3.4 we further assume that there exists a constant k 0 > 0 such that then from (3.5), we infer that the bound from above on v L ∞ (B R ) is independent of R. This fact will have a key role in the proof of global existence for problem (1.1).
Proof of Proposition 3.4. We define where T k (v) has been defined in (3.2) and By (3.6), setting we rewrite 3.7 as Hence, we can apply Lemma 3.3 to v and we obtain Taking the limit as k −→ 0 and we get the thesis.
Let u be the solution to problem (3.2). Then, for a.e. t ∈ (0, T ), where u is the solution to problem (3.2). Observe that Moreover, by direct computation, we get Thus, arguing as in [37,Proposition 2.3], thanks to the comparison principle, we get, for a.e. t ∈ (0, T ), where we have used that T k (u p ) ≤ u p .

L q and smoothing estimates for
Note that the request u ∈ C([0, T ), L q (B R )) for any q ∈ (1, +∞), for any T > 0 is not restrictive, since we will construct solutions belonging to that class (see the proof of Theorem 2.2 below). This remark also applies to several other intermediate results below.
Proof. Since u 0 is bounded and T k is a bounded and Lipschitz function, by standard results, there exists a unique solution of problem (3.2) in the sense of Definition 3.1. We now multiply both sides of the differential equation in problem (3.2) by u q−1 , Now, formally integrating by parts in B R . This can be justified by standard tools, by an approximation procedure. We get Observe that, thanks to Sobolev inequality (1.2), we have Moreover, the last term in the right-hand side of (4.3), thanks to Hölder inequality with exponents N N −2 and N 2 , becomes . (4.5) Combining (4.4) and (4.5), we get . (4.6) Take any T > 0. Observe that, thanks to hypothesis (4.1) and the known continuity of the map t → u(t) in [0, T ], there exists t 0 > 0 such that Hence, (4.6) becomes, for any t ∈ (0, t 0 ], Vol. 21 (2021) where the last inequality is obtained thanks to (4.1). We have proved that t → u(t) L q (B R ) is decreasing in time for any t ∈ (0, t 0 ], i.e. In particular, inequality (4.7) follows for the choice q = p 0 , in view of hypothesis (4.1). Hence, we have Now, we can repeat the same argument in the time interval (t 0 , t 1 ], where t 1 is chosen, due to the continuity of u, in such a way that Thus, we get Iterating this procedure, we obtain that Since T > 0 was arbitrary, the thesis follows.
Using a Moser-type iteration procedure, we prove the following result: . (4.9) Proof. Let {q n } be the sequence defined in (2.7). We start by proving a smoothing estimate from q 0 to qn using a Moser iteration technique (see also [1]). Let t > 0, we define Observe that t 0 = 0, tn = t, {t n } is an increasing sequence w.r.t. n. Now, for any 1 ≤ n ≤ n, we multiply equation (3.2) by u q n−1 −1 and integrate in B R × [t n−1 , t n ]. Thus, we get Then, we integrate by parts in B R × [t n−1 , t n ]. Thanks to Sobolev inequality and hypothesis (4.8), we get dτ, (4.11) where we have used the fact that T k (u p ) ≤ u p . We define q n as in (2.7), so that (m + Hence, in view of hypothesis (4.8) we can apply Lemma 4.1 to the integral on the right-hand side of (4.11), hence we get  Observe that u(·, t n ) (4.13) We define (4.14) By plugging (4.13) and (4.14) into (4.12), we get u(·, t n ) The latter formula can be rewritten as Thanks to the definition of the sequence {q n } in (2.7), we write Observe that, for any 1 ≤ n ≤n, we have for all 1 ≤ n ≤n. Consider the function Observe that, thanks to the definition of σ , g(x) > 0 for any q 0 ≤ x ≤ qn. Moreover, g has a minimum in the interval q 0 ≤ x ≤ qn, call itx. Then, we have for any q 0 ≤ x ≤ qn, x ∈ R. Then, (4.20) becomes We define We have proved a smoothing estimate from q 0 to qn. Observe that if qn = q then the thesis is proved. Now suppose that q > qn. Observe that q 0 ≤ q < qn and define B := N (m − 1)A + 2 q 0 (A + 1).

Remark 4.3.
One cannot let q → +∞ in the above bound. In fact, one can show that ε −→ 0 as q → ∞. So in such limit the hypothesis on the norm of the initial datum (2.2) is satisfied only when u 0 ≡ 0.

Proof of Theorem 2.2
Proof of Theorem 2.2. Let {u 0,h } h≥0 be a sequence of functions such that where p 0 has been defined in (2.1). Observe that, due to assumptions (c) and (d), u 0,h satisfies (2.2). For any R > 0, k > 0, h > 0, consider the problem with s as in (4.26) and γ , δ 1 , δ 2 as in (4.28). In addition, for any τ ∈ (0, T ), ζ ∈ andC > 0 is a constant only depending on m. Inequality (5.5) is formally obtained by multiplying the differential inequality in problem (3.
where all the integrals are finite. Now, observe that, for any h > 0 and R > 0 the sequence of solutions {u R h,k } k≥0 is monotone increasing in k hence it has a pointwise limit for k → ∞. Let u R h be such limit so that we have In view of (5.2), (5.3) and (5.4), the right-hand side of (5.5) is independent of k. So, . We can now pass to the limit as k → +∞ in inequalities (5.2), (5.3) and (5.4) arguing as follows. From inequality (5.2) and (5.3), thanks to the Fatou's Lemma, one has for all t > 0 On the other hand, from (5.4), since u R h,k −→ u R h as k → ∞ pointwise and the right-hand side of (5.4) is independent of k, one has for all t > 0 with s as in (4.26) and γ , δ 1 , δ 2 as in (4.28). Note that (5.7), (5.8) and (5.9) hold for all t > 0, in view of the continuity property of u deduced above. Moreover, thanks to Beppo Levi's monotone convergence theorem, it is possible to compute the limit as k → +∞ in the integrals of equality (5.6) and hence obtain that, for any Observe that all the integrals in (5.10) are finite, hence u R h is a solution to problem (5.1), where we replace T k (u p ) with u p itself, in the sense of Definition 3.1. Indeed, we have, due to (5.7), Now observe that the integral in (5.11) is finite if and only if p γ p < 1 . The latter reads p > m + 2 N , which is guaranteed by the hypotheses of Theorem 2.2. Let us now observe that, for any h > 0, the sequence of solutions {u R h } R>0 is monotone increasing in R, hence it has a pointwise limit as R → +∞. We call its limit function u h so that u R h −→ u h as R → +∞ pointwise. In view of (5.2), (5.3), (5.4), (5.7), (5.8), (5.9), the right-hand side of (5.5) is independent of k and R. So, (u h ) Note that, in view of (5.12), the norms in (5.7), (5.8) and (5.9) do not depend on R (see Lemma 4.1, Proposition 4.2, Proposition 4.4 and Remark 4.5). Therefore, we pass to the limit as R → +∞ in (5.7), (5.8) and (5.9). By Fatou's Lemma, with s as in (4.26) and γ , δ 1 , δ 2 as in (4.28). Note that (5.13), (5.14) and (5.15) hold for all t > 0, in view of the continuity property of u R h deduced above. Moreover, again by monotone convergence, it is possible to compute the limit as R → +∞ in the integrals of equality (5.10) and hence obtain that, for any ϕ ∈ C ∞ c (M × (0, T )) such that ϕ(x, T ) = 0 for any x ∈ M, the function u h satisfies, Observe also that {u h } h≥0 is a monotone increasing function in h hence it has a limit as h → +∞. We call u the limit function. In view (5.2), (5.3), (5.4), (5.7), (5.8), (5.9), (5.13), (5.14) and (5.15) the right-hand side of (5.5) is independent of k, R and h. So, u . Hence, we can pass to the limit as h → +∞ in (5.13), (5.14) and (5.15) and similarly to what we have seen above, we get with s as in (4.26) and γ , δ 1 , δ 2 as in (4.28). Note that both (5.17), (5.18) and (5.19) hold for all t > 0, in view of the continuity property of u deduced above. Moreover, again by monotone convergence, it is possible to compute the limit as h → +∞ in the integrals of equality (5.16) and hence obtain that, for any ϕ ∈ C ∞ c (M × (0, T )) such that ϕ(x, T ) = 0 for any x ∈ M, the function u satisfies, Finally, let us discuss (2.6) and (2.4). Let p 0 ≤ q < ∞, and observe that, thanks to hypotheses (c) and (d), u 0h satisfies hypothesis (2.3) for such q and q 0 = p 0 as u 0 , then we have Hence, due to (5.21), letting k → +∞, R → +∞, h → +∞, by Fatou's Lemma we deduce (2.4).
we choose the sequence u 0h in such a way that it further satisfies u 0,h −→ u 0 in L q (M) as h → +∞ , and observe that u 0h satisfies also (2.5) for such q. Then, we have that Hence, due to (5.22), letting k → +∞, R → +∞, h → +∞, by Fatou's Lemma we deduce (2.6). Then,

Estimates for p > m
Proof. Since u 0 is bounded and T k is a bounded and Lipschitz function, by standard results, there exists a unique solution of problem (3.2) in the sense of Definition 3.1. We now multiply both sides of the differential equation in problem (3.2) by u q−1 , therefore We integrate by parts. This can be justified by standard tools, by an approximation procedure. Using the fact that T (u p ) ≤ u p , we can write . (6.4) Take any α ∈ (0, 1). Thanks to (1.3), (6.4) becomes . (6.5) Moreover, using the interpolation inequality, Hölder inequality and (1.2), we have where θ := m(m+q−1) p( p+q−1) . By plugging (6.5) and (6.6) into (6.3), we obtain Let us now fix α ∈ (0, 1) such that Hence, we have By substituting (6.9) into (6.7), we obtain , (6.10) where C has been defined in Remark 2.6. Observe that, thanks to hypothesis (6.1) and the continuity of the solution u(t), there exists t 0 > 0 such that Hence, (6.10) becomes, for any t ∈ (0, t 0 ] providedε 1 is small enough. Hence, we have proved that u(t) L q (B R ) is decreasing in time for any t ∈ (0, t 0 ], i.e. In particular, inequality (6.11) holds q = p N 2 . Hence, we have Now, we can repeat the same argument in the time interval (t 0 , t 1 ] where t 1 is chosen, thanks to the continuity of u(t), in such a way that Thus, we get Iterating this procedure we obtain the thesis.
Suppose that (2.11) holds for ε 1 = ε 1 ( p, m, N , r, C s , C p ) sufficiently small. Let u be the solution to problem (3.2), such that in addition u ∈ C([0, T ); L q (B R )) for any 1 < q < +∞ and T > 0. Let M support the Sobolev and Poincaré inequalities (1.2) and (1.3). Then, there exists = (N , m, l, C s ) > 0 independent of T such that, for all t > 0, (6.13) Remark 6.3. If in Proposition 6.2, in addition, we assume that for some k 0 > 0 then the bound from above for u(t) L ∞ (B R ) in (6.13) is independent of R.
Proof of Proposition 6.2. Let us set w = u(·, t). Observe that w m ∈ H 1 0 (B R ) and w ≥ 0. Due to Proposition 3.6 we know that Observe that, since u 0 ∈ L ∞ (B R ) also w ∈ L ∞ (B R ). Due to (6.12), we can apply Proposition 3.4, so we have that where s has been defined in (6.12). In view of (2.11) with a suitable ε 1 , since u 0 ∈ L ∞ (B R ), we can apply Lemma 6.1. Hence, we obtain Similarly, again for an appropriate ε 1 in (2.11), since u 0 ∈ L ∞ (B R ), we can apply Lemma 6.1 and obtain Plugging (6.15) and (6.16) into (6.14), we obtain Moreover, since u 0 ∈ L ∞ (B R ), we can apply Lemma 6.1 to w with q = m. Thus, from (6.2) with q = m we get Proof. The proof of Theorem 2.5 follows the same line of arguments of that of Theorem 2.2, with minor differences. Let {u 0,h } h≥0 be a family of functions such that From standard results it follows that problem (6.19) has a solution u R h,k in the sense of Definition 3.1; moreover, u R h,k ∈ C [0, T ]; L q (B R ) for any q > 1. Hence, it satisfies the inequalities in Lemma 6.1 and in Proposition 6.2, i.e. for any t ∈ (0, +∞), with r and s as in (6.12) and as in (6.18). Arguing as in the proof of Theorem (2.6), we can pass to the limit as k → +∞, R → +∞, h → ∞ obtaining a function u, which satisfies  Observe that, due to inequalities (6.20), (6.21) and (6.22), all the integrals in (6.23) are finite, hence u is a solution to problem (1.1) in the sense of Definition 2.1. Finally, using hypothesis (2.12), inequality (2.13) can be derived exactly as (2.6).

Proofs of Theorems 2.8 and 2.9
We use the following Aronson-Benilan-type estimate (see [2]; see also [37, Proposition 2.3]); it can be shown exactly as Proposition 3.6. For any R > 0, consider the following approximate problem where B R denotes the Euclidean ball with radius R and centre in the origin O.
We exploit the following estimate, which can be proved as that in Lemma 4.1.

Lemma 7.2. Let
Suppose that inequality (1.5) holds. Suppose that u 0 ∈ L ∞ (B R ), u 0 ≥ 0. Let 1 < q < ∞, p 0 be as in (2.1) and assume that u 0 L p 0 ρ (B R ) <ε, forε =ε( p, m, C s , q) small enough. Let u be the solution of problem (7.1), such that in addition u ∈ C([0, T ), L q ρ (B R )) for any q ∈ (1, +∞), for any T > 0. Then, The following smoothing estimate is also used; the proof is the same as that of Proposition 4.2. ; δ q = q 0 q q + N 2 (m − 1) q 0 + N 2 (m − 1) . We consider problem (7.1). We use the following estimate, which can be proved as that in Lemma 6.1. Proof of Theorem 2.9. The conclusion follows arguing step by step as in the proof of Theorem 2.5. We use Lemma 7.4 instead of Lemma 6.1 and Proposition 7.1 instead of Proposition 3.6.