Nonlinear evolution problems with singular coefficients in the lower order terms

We consider a Cauchy Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite time horizon.

For the notation related to parabolic type function spaces such as L p 0, T, W 1,p 0 (Ω) or similar, we refer the reader to Section 2 below. Above and throughout the paper, for two vectors ξ, η ∈ R N we denote by ξ · η their scalar product and we denote by ·, · the duality between W −1,p ′ (Ω) and W 1,p 0 (Ω). Model equation that we consider in this context is (1.9) u t − div |∇u| p−2 ∇u + |u| p−2 u µ h(t) |x| where µ > 0, h ∈ L ∞ (0, T ) and b 0 ∈ L ∞ (Ω T ).
When p = 2, the linear homogeneous equation in (1.1) describes the evolution of some Brownian motion and it is also known as Fokker-Planck equation (see e.g. [3,17] and the references therein). We remark that the boundedness of the growth coefficient b(x, t) is too restrictive in many applications as, for instance, in the case of the diffusion model for semiconductor devices (see e.g. [4]). On the other hand, a low integrability assumption in Ω T for the term b = b(x, t) does not guarantee the existence of a distributional solution in the sense of definition (1.8). In this case, other definitions of solutions have been introduced (see e.g. [17]). Assumption (1.7), in view of Sobolev embedding theorem (see Theorem 2.1 below), guarantees that Ω T A(x, t, u, ∇u) · ∇u dx dt < ∞ that is, a solution in the sense of Definition 1 has finite energy.
Our existence result reads as follows.
Then problem (1.1) admits a solution.
The constant S N,p is the one of Sobolev embedding theorem in Lorentz spaces (see Theorem 2.1 below). In (1.10) dist L N,∞ (Ω) (b, L ∞ (Ω)) denotes the distance from bounded functions of the function b with respect to the L N,∞ -norm (see formula (2.3) below for the definition).
Condition (1.10) , for the first time introduced in [10] and in [11], does not imply a smallness condition on the norm. Indeed, in the example (1.9) it just gives a bound on the constant µ. In particular, (1.10) holds true whenever b ∈ L ∞ 0, T, L N,q (Ω) for 1 q < ∞. Here L N,q (Ω) denotes the Lorentz space (see Section 2.2 for the definition). We don't know if condition (1.10) is optimal in our framework. Nevertheless, in the elliptic counterpart of Theorem 1.1 (considered in [7]) such a condition turns to be optimal at least for p = 2.
We also study the behavior on time of a weak solution given in Theorem 1.1. More precisely, we estimate on time the L 2 -norm of u with the solution of a Cauchy problem related to a o.d.e. (see Theorem 6.1). As consequence we provide estimates that highlight the different decay behavior as the exponent p varies when T = ∞. The presence of the lower order term does not affect the decay to zero of the L 2 -norm as T goes to infinity (Corollary 6.2 below). Recent results about the time decay for solutions to parabolic problems in absence of the lower order term can be found in [8,14,18] and the references therein.
The novelty of the paper lies on the fact that in Theorem 1.1 above and Theorem 6.1 below we consider a family of operators not coercive with a singular growth coefficient in the lower order term. We recall that bounded functions are not dense in the Marcinkiewicz space L N,∞ (Ω). In order to find a solution to (1.1), the main tool is an apriori estimate that could have interest by itself (see Proposition 3.2). Thanks to Leray-Schauder fixed point theorem, we first solve the problem when b(x, t) is bounded. Then, we obtain a solution to (1.1) as a limit of a sequence of solutions to suitable approximating problems. A solution of Theorem 1.1 satisfies an energy equality and then by using a recent result of [8] we are able to describe its asymptotic behavior.

Preliminary results
2.1. Basic notation. We adopt the usual symbols and for inequalities which holds true up to not influent constants (in our case typically depending on N, p, α and β).
We will denote by C (or by similar symbols such as C 1 , C 2 , . . . ) a generic positive constant, which may possibly vary from line to line. The dependence of C upon various parameters will be highlighted in parentheses, adopting a notation of the type C(·, . . . , ·).

Function spaces.
Let Ω be a bounded domain in R N . Given 1 < p < ∞ and 1 q < ∞, the Lorentz space L p,q (Ω) consists of all measurable functions f defined on Ω for which the quantity is finite, where λ f (k) := |{x ∈ Ω : |f (x)| > k}| is the distribution function of f . Note that · p,q is equivalent to a norm and L p,q (Ω) becomes a Banach space when endowed with it (see [2,16]). For p = q, the Lorentz space L p,p (Ω) reduces to the Lebesgue space L p (Ω). For q = ∞, the class L p,∞ (Ω) consists of all measurable functions f defined on Ω such that and it coincides with the Marcinkiewicz class, weak-L p (Ω). For Lorentz spaces the following inclusions hold whenever 1 q < p < r ∞. Moreover, for 1 < p < ∞, 1 q ∞ and 1 p + 1 p ′ = 1, 1 q + 1 q ′ = 1, if f ∈ L p,q (Ω), g ∈ L p ′ ,q ′ (Ω) we have the Hölder-type inequality As it is well known, L ∞ (Ω) is not dense in L p,∞ (Ω). For a function f ∈ L p,∞ (Ω) we define In order to characterize the distance in (1.10), we introduce for all k > 0 the truncation operator at heights k, namely T k y := y |y| min{|y|, k} for y ∈ R .
It is easy to verify that Clearly, for 1 q < ∞ any function in L p,q (Ω) has vanishing distance to L ∞ (Ω). Indeed, L ∞ (Ω) is dense in L p,q (Ω), the latter being continuously embedded into L p,∞ (Ω).
Assuming that 0 ∈ Ω, a typical element of L N,∞ (Ω) is b(x) = B/|x|, with B a positive constant. An elementary calculation shows that (2.5) dist where ω N stands for the Lebesgue measure of the unit ball of R N .

EVOLUTION PROBLEMS WITH SINGULAR COEFFICIENTS
The Sobolev embedding theorem in Lorentz spaces [16,1] reads as Theorem 2.1. Let us assume that 1 < p < N, 1 q p, then every function u ∈ W 1,1 0 (Ω) verifying |∇u| ∈ L p,q (Ω) actually belongs to L p * ,q (Ω), where p * := N p N −p is the Sobolev exponent of p and For our purposes, we also need to introduce some spaces involving the time variable. Hereafter, for the time derivative u t of a function u we adopt the alternative notation ∂ t u,u, u ′ or du/ dt. Let T > 0. If we let X be a separable Banach space endowed with a norm · X , the space L p (0, T, X) is defined as the class of all measurable functions u : We will mainly deal with the case where X is either a Sobolev space or a Lorentz space. In particular, we recall a well known result (see e.g. [19, Proposition 1.2, Chapter III, pag. 106]) involving the class of functions W p (0, T ) defined as follows Finally, we recall the classical compactness result due to Aubin-Lions (see e.g. [19, Proposition 1.3, Chapter III, pag. 106]).

Lemma 2.3 (Aubin-Lions)
. Let X 0 , X, X 1 be Banach spaces with X 0 and X 1 reflexive. Assume that X 0 is compactly embedded into X and X is continuosly embedded into X 1 .
Then W is compactly embedded into L p (0, T, X).
A prototypical example of application of this lemma corresponds to the choices q = p, Obviously L 2 (Ω) ⊂ L p (Ω) as long as p < 2, and therefore we deduce the following.
We recall a couple of lemmas (see [8]) whose application will be essential in the study of the time behavior of the solutions to (1.1).
Let g ∈ L 1 (t 0 , T ; R) and γ : [t 0 , T ] → R + be measurable, bounded and satisfy Then there exists a solution x(·) ∈ W 1,1 ([t 0 , T ]) of the Cauchy problem , +∞) → R + is measurable and locally bounded and for every T > t 0 the above assumptions hold true, then there exists a solution x to (2.9) defined on [t 0 , ∞) such that γ ≤ x. In particular, Lemma 2.6. Under all the assumptions of Lemma 2.5 suppose that ψ(t, a) = 0 for all a ≤ 0, g(·) ≥ 0, ψ(t, ·) is increasing for a.e. t ∈ [t 0 , T ] and that for any R > r > 0 there exists Then the solution z(·) of is unique and well defined on [t 0 , T ], z(·) ≥ 0 and γ(t) ≤ x(t) ≤ z(t) +

Weak type and a priori estimates for an auxiliary problem
An a priori estimate on the distribution function of a solution to problem (1.1) will be fundamental in order to prove Theorem 1.1. To this aim, we let φ : R → R be the function defined as We set Φ(w) := |w| 0 φ(ρ) dρ and Ψ : (0, ∞) → (0, ∞) be the reciprocal of the restriction of Φ to (0, ∞), so that it is a continuous and decreasing function such that Ψ(k) → 0 as k → ∞.
With this notation at hand, we have the following result.
Proof. First of all, we set w := u/λ, in such a way that w solves the problem We fix t ∈ (0, T ) and we choose ϕ := φ(w)χ (0,t) as a test function for (3.4). (This can be done since φ(·) is a Lipschitz function in the whole of R.) In this way, we have Observe explicitly that By means of Young inequality we have In view of the latter estimate and of the fact that 0 Φ(k) 1 2 k 2 for any k 0, from (3.6) we infer Proof. We set w := u/λ, in such a way that w solves the problem (3.4). We fix t ∈ (0, T ) and we choose ϕ := wχ (0,t) as a test function for (3.4) so we get By means of Young inequality we have for 0 < ε < 1 Now, we provide an estimate on λb|w| L p (Ωt) = b|u| L p (Ωt) . By Minkowski inequality we have

EVOLUTION PROBLEMS WITH SINGULAR COEFFICIENTS
Here T m b denotes the truncation b at levels ±m. We estimate separately the latter two terms. For k > 0 fixed, we have In particular, we apply Hölder inequality (2.2), Sobolev inequality (2.6) slice-wise and Lemma 3.1 to get We are able to reabsorb by the left hand side by choosing properly m, k and ε. For instance, it is sufficient to have m so large to guarantee and the existence of such value of m is a direct consequence of (1.10). A proper choice of k and ε can be performed coherently with (3.19) and taking into account the properties of Ψ(·). In particular, denoting by C a constant depending only on N, p, α and on Taking into account (3.20) and recalling that λ ∈ (0, 1], the latter estimate leads to the conclusion of the proof. Remark 3.3. We point out that the a priori estimate (3.9) is uniform with respect to the parameter λ.

Parabolic equations with bounded coefficients
This section is devoted to the proof of the existence of a solution to problem (1.1) in the special case b ∈ L ∞ (Ω T ). We shall use on this account the following version of Leray-Schauder fixed point theorem as in (see e.g. [9, Theorem 11.3 pg. 280]).
Theorem 4.1. Let F be a compact mapping of a Banach space X into itself, and suppose there exists a constant M such that x X < M for all x ∈ X and λ ∈ [0, 1] satisfying x = λF (x). Then, F has a fixed point.
We recall that a continuous mapping between two Banach spaces is called compact if the images of bounded sets are precompact.
Accordingly, the main result of this section reads as follows. Proof. We let v ∈ L p (Ω T ), and consider the problem Problem (4.1) admits a solution by the classical theory of pseudomonotone operators ( [12]) and by the strict monotonicity of the vector field (x, t, ξ) ∈ Ω T × R N →Ā(x, t, ξ) := A(x, t, v, ξ) such a solution is unique. Hence, the map F which takes v to the solution u is well defined and certainly acts from L p (Ω T ) into itself. Our goal is to determine a fixed point for F , which is obviously a solution to (1.1) under the assumption b ∈ L ∞ (Ω T ). We want to apply Theorem 4.1, so we need to show that F : L p (Ω T ) → L p (Ω T ) is continuous, compact and the set . First observe that the boundedness of U is a direct consequence of Proposition 3.2.
Let us prove that {F [v n ]} n∈N is a precompact sequence if {v n } n∈N is a bounded sequence in L p (Ω T ). We need to show that u n := F [v n ] admits a subsequence strongly converging in L p (Ω T ). By definition of F , we see that u n solves the problem in Ω T , u n = 0 on ∂Ω × (0, T ), u n (·, 0) = u 0 in Ω, By testing the equation in (4.2) by u n itself and arguing similarly as before, we see that So in particular {|∇u n |} n∈N is bounded in L p (Ω T ). Using the equation in (4.1) we see that {∂ t u n } n∈N is bounded in L p ′ 0, T, W −1,p ′ (Ω) and so {u n } n∈N strongly converges in L p (Ω T ) as a direct consequence of the Aubin-Lions lemma.
Let us prove the continuity of F . Let {v n } n∈N be a strongly converging sequence in L p (Ω T ), say v n → v in L p (Ω T ) strongly We set u n := F [v n ]. We already know that {u n } n∈N is compact sequence in L p (Ω T ) and also that estimate (4.3) holds true. So, there exists u ∈ W p (0, T ) such that Observe further that u ∈ C 0 ([0, T ], L 2 (Ω)) with u(·, 0) = u 0 . It is a direct consequence of the inclusion of Lemma 2.2, the boundednes of {u n } n∈N in W p (0, T ) and of the convergence u n ⇀ u weakly in L 2 (Ω) for all t ∈ [0, T ]. In order to prove the continuity of F , we need to show that Again, we know that u n solves (4.2), namely for every ϕ ∈ C ∞ (Ω T ) with support contained in [0, T ) × Ω we have If we choose u n − u as a test function in the above identity we have A(x, t, v n , ∇u n ) · ∇(u n − u) dx dt = 0 (4.12) It is clear that {A(x, s, v n , ∇u n )} n∈N is bounded in L p ′ (Ω T ) and so it weakly converges in L p ′ (Ω T ) to someÃ. We use the Minty trick to recover thatÃ(x, t) = A(x, t, v(x, t), ∇u(x, t)) a.e. in Ω T . Namely, let η ∈ L p (Ω T , R N ). Observe that Passing to the limit, we get We choose η := ∇u − λψ in (4.14) where ψ ∈ L p (Ω T , R N ) and λ ∈ R. Then If we assume that λ > 0, we divide by λ itself and then letting λ → 0 + we have Arguing similarly if λ < 0 we get the opposite inequality than (4.16), so we conclude that for every ψ ∈ L p (Ω T , R N ) i.e.Ã(x, t) = A(x, t, v(x, t), ∇u(x, t)) a.e. in Ω T .
We are in position to pass to the limit in (4.9). Therefore

Proof of the existence result via approximation scheme
Proof of Theorem 1.1. Let n ∈ N. We introduce the following initial-boundary value problem To achieve the proof of Theorem 1.1 we need to pass to the limit in (5.1). The results of Section 4 provides the existence of solution u n ∈ C 0 [0, T ], L 2 (Ω) ∩ L p 0, T, W 1,p 0 (Ω) to problem (5.1). In fact, A n = A n (x, t, u, ξ) satisfies (1.4), (1.5) and (1.6) with T n b in place of b and T n b belongs to L ∞ (Ω T ) for each fixed n ∈ N. Since T n b b in Ω T for every n ∈ N, by Proposition 3.2, there exists a positive constant independent of n such that the following estimate for a solution to problem (5.1) holds sup 0<t<T Ω |u n (·, t)| 2 dx + Ω T |∇u n | p dx dt C (5.2) Hence, there exists u ∈ L ∞ (0, T, L 2 (Ω)) ∩ L p 0, T, W 1,p 0 (Ω) such that u n ⇀ u weakly in L p (Ω T ) (5.3) ∇u n ⇀ ∇u weakly in L p Ω T , R N (5.4) u n ⇀ * u weakly * in L ∞ (0, T ; L 2 (Ω)) (5.5) as n → ∞. With the aid of the equation in (5.1), we obtain a uniform bound for the norm of the time derivative of u n in L p ′ 0, T, W −1,p ′ (Ω) . Therefore, by Aubin-Lions lemma we have u n → u strongly in L p (Ω T ) and a.e. in Ω T (5.6) Note also that u ∈ C 0 ([0, T ], L 2 (Ω)) and u(·, 0) = u 0 . As before, this is a consequence of Lemma 2.2, the boundednes of {u n } n∈N in W p (0, T ) and of the convergence u n ⇀ u weakly in L 2 (Ω) for all t ∈ [0, T ].
In view of (5.4), to get (5.14) it suffices to show that Preliminarily, we observe that combining (5.6) with the property that θ n → 1 as n → ∞, we have A n (x, t, u n , ∇u) 1 + |u n − u| p → A(x, t, u, ∇u) a.e. in Ω .
On the other hand, we see that Now, from the monotonicity condition (1.5), (5.13) and (5.14) we get As the integrand is nonnegative, we have (up to a subsequence) [A n (x, t, u n , ∇u n ) − A n (x, t, u n , ∇u)] ∇γ(u n − u) → 0 a.e. in Ω T .
Arguing as in the proof of [13,Lemma 3.3], we see that (5.18) ∇u n → ∇u a.e. in Ω T

and (5.19)
A n (x, t, u n , ∇u n ) ⇀ A(x, t, u, ∇u) in L p ′ (Ω T , R N ) weakly and we conclude that u solves the original problem (1.1).
We conclude this section providing an example which shows that assumption (1.10) in general cannot be dropped, even for linear problems.

Asymptotic behavior
This section is devoted to study the time behavior of a solution to problem (1.1). Through this section we assume that (1.4), (1.6) and (1.7) are in charge and that condition (1.10) is satisfied.
For convenience we will denote by for a.e. t ∈ [0, T ]. The first result of the present section is the following Theorem 6.1. Under the above assumptions, any solution u to problem (1.1) satisfies for any t ∈ [0, T ] the following estimate where x(t) is the unique solution of the problem Here C > 0 and M i > 0, i = 1, 2, are positive constants depending on N, p, α, β, |Ω| and D b .
Proof. We test the equation in (1.1) by the solution u itself. We get the following energy We write down such equality first for t = t 1 and subsequently for t = t 2 , we subtract the relations obtained in this way and we deduce A(x, s, u, ∇u) · ∇u dx ds = Using (1.4) we get H(x, s) dx ds + By Young inequality the latter inequality implies From Proposition 3.2 (applied for λ = 1) it is clear that ∇u L p (Ω T ) is controlled by a quantity depending only on the data, so Hölder and Sobolev inequalities applied slice-wise give us Taking into accout all the above relations, finally we get for some constants C, C 0 depending on the data. By Sobolev inequality we have Since p > 2N/(N + 2) is equivalent to p * > 2, then − Ω |u(·, s)| p * dx for a.e. s ∈ (t 1 , t 2 ). Hence, from (6.7) and therefore from (6.8) g(s) ds (6.10) Finally, our claimed result follows directly by Lemma 2.5 where we choose t 0 = 0, φ(t) := u(t) 2 L 2 (Ω) and ψ(t, y) ≡ ψ(y) := M|y| 1+ν for M positive constant and ν = p/2 − 1 for p > 2 and ν = 0 for 2N/(N + 2) < p 2.
As a byproduct of previous result, we are able to show that the L 2 (Ω)-norm of any solution to problem (1.1) decays as an explicit negative power of the time variable and exponentially fast respectively in case p > 2 and 2N/(N + 2) < p 2 by using Lemma 2.6. The proof follows the same lines as in Proposition 3.1 and Proposition 3.2 of [14].
Remark 6.4. We remark that when p > 2 previous proposition provides universal estimate on time, in the sense that the estimate does not depend on the initial datum u 0 .
Remark 6.5. The statement of Theorem 6.1 improves the behavior in time of a solution to problem (1.1) known so far [8,14] for p-Laplace operator and see also [5,6] for p = 2.